arxiv:2002.06839v1 [math-ph] 17 feb 2020

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Integrable models and K-theoretic pushforward

of Grothendieck classes

Kohei Motegi ∗

Faculty of Marine Technology, Tokyo University of Marine Science and Technology,

Etchujima 2-1-6, Koto-Ku, Tokyo, 135-8533, Japan

August 4, 2021

Abstract

We show that a multiple commutation relation of the Yang-Baxter algebra of inte-grable lattice models derived by Shigechi and Uchiyama can be used to connect twotypes of Grothendieck classes by the K-theoretic pushforward from the Grothendieckgroup of Grassmann bundles to the Grothendieck group of a nonsingular variety. Usingthe commutation relation, we show that two types of partition functions of an integrablefive-vertex model, which can be explicitly described using skew Grothendieck polynomi-als, and can be viewed as Grothendieck classes, are directly connected by the K-theoreticpushforward. We show that special cases of the pushforward formula which correspondto the nonskew version are also special cases of the formulas derived by Buch. We alsopresent a skew generalization of an identity for the Grothendieck polynomials by Guoand Sun, which is an extension of the one for Schur polynomials by Feher, Nemethiand Rimanyi. We also show an application of the pushforward formula and derive anintegration formula for the Grothendieck polynomials.

1 Introduction

Recently, various connections between integrable systems [1, 2, 3] and K-theoretic objects inalgebraic geometry have been explored [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. A typical objectwhich appears in both areas is the Grothendieck polynomials [15, 16, 17, 18, 19, 20, 21] whichare polynomial representatives for K-theoretic Schubert classes in algebraic geometry. Theyappear in integrable models as explicit forms of partition functions in statistical physics.The study of partition functions of integrable lattice models based on the quantum inversescattering method [3, 22, 23] was initiated by Korepin [24] and Izergin [25]. See [26, 27,28, 29, 30, 31] for examples on the developments of the method. The partition functionsare found to be connected with famous symmetric functions and their generalizations, and

∗E-mail: [email protected]

1

http://arxiv.org/abs/2002.06839v2

based on the correspondence between partition functions and symmetric functions, variousalgebraic identities have been found. See the aforementioned papers as well as [32, 33, 34, 35,36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50] for examples on the developments onthis subject. As for the Grothendieck polynomials, the connection with quantum integrablemodels gave new perspectives on algebraic identities and also on the Littlewood-Richardsoncoefficients, Gromov-Witten invariants and so on [4, 5, 9, 10].

In this paper, we discuss another aspect of the connection between integrable modelsand algebraic geometry. We investigate the K-theoretic pushforward from the Grothendieckgroup of Grassmann bundles to the Grothendieck group of a nonsingular variety from theperspective of quantum integrability.

Let us first recall the algebraic geometry side. See [17, 18, 51, 52] for examples for moredetails. Let X be a smooth projective complex variety and K(X) the Grothendieck ringof isomorphism classes of algebraic vector bundles over X. For a subvariety Y of X, theGrothendieck class of the structure sheaf of Y is denoted [OY ], or [Y ] for short. Let A → Xbe a rank n vector bundle, and π : Grk(A) → X the Grassmann bundle of k-planes in Awith universal tautological exact sequence 0 → S → π∗A → Q → 0 of vector bundles overGrk(A). Let σ = {σ1, . . . , σk} and ω = {ω1, . . . , ωn−k} be the set of Grothendieck roots forthe subbundle S and the quotient bundle Q respectively, i.e., defined by the exterior powersof S and Q by the relation [

∧i S] = ei(σ) and [∧j Q] = ej(ω), where ei(σ) and ej(ω) are the

ith and jth elementary symmetric functions with the set of variables σ and ω, respectively.Let f(σ;ω) be a Laurent polynomial in K(X)[σ±

i ;ω±j ] which are symmetric in σ variables

and also symmetric in ω variables, which makes it possible for the polynomial to be viewedas a certain Grothendieck class in K(Grk(A)).

We consider the following K-theoretic pushforward π∗ : K(Grk(A)) → K(X) inducedfrom π. The following localization type formula holds for π∗(f) [51, 52, 53, 54, 55, 56, 57, 58,59, 60]:

Proposition 1.1. Let Snk be a k-subset of [n] = {1, 2, . . . , n} and denote the set of k-subsets

of n as([n]k

), and Sn

k = {1, 2, . . . , n}\Snk . Let {α1, . . . , αn} be the Grothendieck roots for A

and αJ = {αj1 , . . . , αjr} for any subset J = {j1, . . . , jr} of [n]. For a Grothendieck classf(σ;ω) ∈ K(Grk(A)), we have

π∗(f(σ;ω)) =∑

Snk∈([n]

k )

∏

i∈Snk,j∈Sn

k

1

1− αi/αjf(αSn

k;αSn

k). (1.1)

The explicit computations of the pushforward have been explored through various ways.One of the approaches which have been studied extensively in recent years is to reexpressthe right hand side of (1.1) as iterated residues and find various pushforward formulas forpolynomials which can be viewed as generalized cohomology classes [51, 52, 61, 62, 63, 64,65, 66, 67, 68, 69, 70, 71, 72, 73, 74].

In this paper, we study the map π∗ of Grothendieck classes from the point of view ofquantum integrability. On the integrable model side, the key observation is that the righthand side of the following commutation relation between the monodromy matrix elements of

2

certain integrable models

n∏

j=k+1

D(uj)k∏

j=1

B(uj) =∑

Snk∈([n]

k )

∏

i∈Snk,j∈Sn

k

1

1− ui/uj

∏

i∈Snk

B(ui)∏

j∈Snk

D(uj), (1.2)

whose details of the notations will be explained later, resembles the right hand side of theK-theoretic pushforward formula (1.1). This type of commutation relation is one of thecommutation relations derived essentially by Shigechi and Uchiyama [35] when they inves-tigated the integrable boson model [34], and also holds for an integrable five vertex modelused in this paper. One can observe that the the expressions in the right hand side of (1.1)and (1.2) resemble. However, in (1.2), what appears are the so-called B-operators and D-operators which are noncommutative objects. Noncommutative objects do not appear inthe K-theoretic pushforward formula (1.1). To go from the noncommutative world to thecommutative world, we consider partition functions of an integrable five-vertex model, whichare fundamental objects in statistical physics. We introduce two classes of partition func-tions which can be regarded as some Grothendieck classes of the Grassmann bundle Grk(A)and a nonsingular variety X. Using the Shigechi-Uchiyama commutation relation (1.2), weshow that they are directly connected by the K-theoretic pushforward π∗. We also showthat the partition functions introduced in this paper can be explicitly described using theskew Grothendieck polynomials, i.e., we show that two polynomials described by the skewGrothendieck polynomials are connected by the K-theoretic pushforward. It is also shownthat special cases of the formula which correspond to the nonskew version are also special casesof the formulas derived by Buch [17]. We also present an identity for the skew Grothendieckpolynomials by taking matrix elements of the commutation relation (1.2), which can be re-garded as a skew extension of the identity for the Grothendieck polynomials by Guo and Sun[75], which is an extension of an identity for the Schur polynomials by Feher, Nemethi andRimanyi [76]. By combining the pushforward formula with another approach, we also get anintegration formula for the Grothendieck polynomials. It would be interesting to investigateother kinds of maps which appear in algebraic geometry from the point of view of quantumintegrability by observing similarities between formulas appearing in both areas.

This paper is organized as follows. In the next section, we introduce an integrable five-vertex model and the Yang-Baxter algebra associated with the model, and review a com-mutation relation by Shigechi and Uchiyama. In section 3, we introduce partition functionsconstructed from the R-matrix of the integrable five-vertex model, and write their explicitforms using skew Grothendieck polynomials. In section 4, we prove a correspondence formulabetween two types of Grothendieck classes of the Grassmann bundle Grk(A) and a nonsin-gular variety X by the K-theoretic pushforward. The proof is given by using the partitionfunction represenations of the Grothendieck classes and the Shigechi-Uchiyama commutationrelation. We also present an identity for the skew Grothendieck polynomials obtained as thematrix elements of the commutation relation. We show in section 5 that special cases of theformula and identity are also special cases of the formulas derived by Buch and Guo-Sun.We also show an application of the pushforward formula and derive an integration formulafor the Grothendieck polynomials.

3

2 Integrable five-vertex model

In this section, we introduce an integrable five-vertex model and the Yang-Baxter algebraassociated with the model.

We first introduce a two-dimensional vector space W , and denote its standard basis as

{|0〉, |1〉}, where |0〉 :=

(10

), |1〉 :=

(01

). We denote the dual of |0〉, |1〉 as 〈0| := (1, 0),

〈1| := (0, 1), and the dual vector space as W ∗. We use the bra-ket notation (without taking

complex conjugation) in this paper. We define 〈A|B〉 for two vectors |A〉 =

(A1

A2

), |B〉 =

(B1

B2

), A1, A2, B1, B2 ∈ C as the standard inner product between |A〉 ∈ W and |B〉 ∈ W :

〈A|B〉 := A · B = A1B1 + A2B2, and view this as the product between 〈A| and |B〉 where〈A| := (A1, A2) ∈ W ∗. The inner product between basis vectors (product between basis anddual basis vectors) are 〈0|0〉 = 〈1|1〉 = 1, 〈1|0〉 = 〈0|1〉 = 0. Using the bra-ket notation,the identity operator Id acting on W can be written as Id = |0〉〈0| + |1〉〈1|. This can bechecked by noting that the action of |0〉〈0| + |1〉〈1| on |0〉 and |1〉 are (|0〉〈0| + |1〉〈1|)|0〉 =|0〉〈0|0〉 + |1〉〈1|0〉 = |0〉 and (|0〉〈0| + |1〉〈1|)|1〉 = |0〉〈0|1〉 + |1〉〈1|1〉 = |1〉, and any vector inW can be written as a linear combination of |0〉 and |1〉.

Next, we consider the tensor product of the two-dimensional vector spaces Wi ⊗ Wj .Here we introduce subscripts to distinguish the two-dimensional vector spaces. We can take{|0〉i ⊗ |0〉j , |0〉i ⊗ |1〉j , |1〉i ⊗ |0〉j , |1〉i ⊗ |1〉j} as a basis of Wi ⊗Wj . The R-matrix Rij(u,w)acting on Wi ⊗Wj is defined by acting on this basis as

Rij(u,w)|0〉i ⊗ |0〉j = |0〉i ⊗ |0〉j , (2.1)

Rij(u,w)|0〉i ⊗ |1〉j = w/u|1〉i ⊗ |0〉j , (2.2)

Rij(u,w)|1〉i ⊗ |0〉j = |0〉i ⊗ |1〉j + (1− w/u)|1〉i ⊗ |0〉j , (2.3)

Rij(u,w)|1〉i ⊗ |1〉j = |1〉i ⊗ |1〉j . (2.4)

Here, u and w are complex numbers. To indicate the vector spaces which the R-matrix act,we use subscripts and denote the R-matrix as Rij(u,w).

We also introduce the following notation for the matrix elements: we define [R(u,w)]δ1δ2ǫ1ǫ2

for ǫ1, ǫ2, δ1, δ2 = 0, 1 as Rij(u,w)|ǫ1〉i ⊗ |ǫ2〉j =∑

δ1,δ2=0,1

[R(u,w)]δ1δ2ǫ1ǫ2 |δ1〉i ⊗ |δ2〉j . By defintion

of the R-matrix (2.1), (2.2), (2.3) and (2.4), we have [R(u,w)]0000 = [R(u,w)]1111 = [R(u,w)]0110 =1, [R(u,w)]1001 = w/u, [R(u,w)]1010 = 1− w/u and [R(u,w)]δ1δ2ǫ1ǫ2 = 0 otherwise. An importantproperty of the matrix elements for this R-matrix is [R(u,w)]δ1δ2ǫ1ǫ2 = 0 if ǫ1 + ǫ2 6= δ1 + δ2.This property is called as the ice-rule. We also remark that [R(u,w)]δ1δ2ǫ1ǫ2 is the same with

i〈δ1| ⊗ j〈δ2|Rij(u,w)|ǫ1〉i ⊗ |ǫ2〉j using the bra-ket notation.

Identifying |0〉i⊗|0〉j , |0〉i⊗|1〉j , |1〉i⊗|0〉j , |1〉i⊗|1〉j , with

1000

,

0100

,

0010

,

0001

,

4

respectively, the R-matrix is written as the 4× 4 matrix

Rij(u,w) =

1 0 0 00 0 1 00 w/u 1− w/u 00 0 0 1

. (2.5)

We remark that the R-matrix (2.5) can be regarded as a certain limit of the Uq(sl2) R-matrix[77, 78, 79] for the six-vertex model. See Figure 1 for a graphical description of the R-matrixof the integrable five-vertex model.

More generally, for an integer m ≥ 2, we define Rij(u,w) (1 ≤ i < j ≤ m) as an operatoracting on W1⊗W2⊗· · ·⊗Wm. We can take {|ǫ1〉1⊗|ǫ2〉2⊗· · ·⊗ |ǫm〉m | ǫ1, ǫ2, . . . , ǫm = 0, 1}as a basis of W1 ⊗W2 ⊗ · · · ⊗Wm. The operator Rij(u,w) is defined by acting on this basisas

Rij(u,w)|ǫ1〉1 ⊗ · · · ⊗ |0〉i ⊗ · · · ⊗ |0〉j ⊗ · · · ⊗ |ǫm〉m

= |ǫ1〉1 ⊗ · · · ⊗ |0〉i ⊗ · · · ⊗ |0〉j ⊗ · · · ⊗ |ǫm〉m, (2.6)

Rij(u,w)|ǫ1〉1 ⊗ · · · ⊗ |0〉i ⊗ · · · ⊗ |1〉j ⊗ · · · ⊗ |ǫm〉m

= w/u|ǫ1〉1 ⊗ · · · ⊗ |1〉i ⊗ · · · ⊗ |0〉j ⊗ · · · ⊗ |ǫm〉m, (2.7)

Rij(u,w)|ǫ1〉1 ⊗ · · · ⊗ |1〉i ⊗ · · · ⊗ |0〉j ⊗ · · · ⊗ |ǫm〉m

= |ǫ1〉1 ⊗ · · · ⊗ |0〉i ⊗ · · · ⊗ |1〉j ⊗ · · · ⊗ |ǫm〉m

+ (1− w/u)|ǫ1〉1 ⊗ · · · ⊗ |1〉i ⊗ · · · ⊗ |0〉j ⊗ · · · ⊗ |ǫm〉m, (2.8)

Rij(u,w)|ǫ1〉1 ⊗ · · · ⊗ |1〉i ⊗ · · · ⊗ |1〉j ⊗ · · · ⊗ |ǫm〉m

= |ǫ1〉1 ⊗ · · · ⊗ |1〉i ⊗ · · · ⊗ |1〉j ⊗ · · · ⊗ |ǫm〉m, (2.9)

for any ǫk = 0, 1 (k = 1, . . . ,m, k 6= i, j). The subscripts i and j of Rij(u,w) indicate thevector spaces which the operator acts nontrivially. Note that Rij(u,w) acts as identity onvector spaces except for the spaces Wi and Wj, and the action (2.6), (2.7), (2.8) and (2.9) canbe regarded as an extension of the action (2.1), (2.2), (2.3) and (2.4) to the space W1 ⊗ · · · ⊗Wm. Rij(u,w) acts on the 2m-dimensional vector space W1⊗· · ·⊗Wm, hence can be regardedas a 2m × 2m matrix. The matrix elements [Rij(u,w)]

δ1 ···δmǫ1···ǫm for ǫ1, . . . , ǫm, δ1, · · · , δm = 0, 1

defined as Rij(u,w)|ǫ1〉1 ⊗ · · · ⊗ |ǫm〉m =∑

δ1,...,δm=0,1

[Rij(u,w)]δ1 ···δmǫ1···ǫm |δ1〉1 ⊗ · · · ⊗ |δm〉m can be

expressed using [R(u,w)]δiδjǫiǫj as

[Rij(u,w)]δ1 ···δmǫ1···ǫm = [R(u,w)]

δiδjǫiǫj

m∏

k=1k 6=i,j

δǫkδk , (2.10)

where δǫkδk = 1 if ǫk = δk, and δǫkδk = 0 otherwise.The operator Rij(u,w) defined as (2.6), (2.7), (2.8), (2.9) satisifies the Yang-Baxter rela-

tion (Figure 2)

Rij(u, v)Rik(u,w)Rjk(v,w) = Rjk(v,w)Rik(u,w)Rij(u, v), (2.11)

5

Figure 1: A graphical description of the R-matrix Rij(u,w) of the integrable five-vertexmodel (2.5). The elements of the R-matrix are displayed. The R-matrix is represented as twocrossing arrows. The left and the up arrow represents the space Wi and Wj, respectively. 0or 1 on the left and the right around a vertex denote the input (basis vector) and the output(dual basis vector) of Wi, and 0 or 1 on the bottom and the top denote the input (basisvector) and the output (dual basis vector) of Wj, respectively. For example, [Rij(u,w)]

1001 =

i〈1| ⊗ j〈0|Rij(u,w)|0〉i ⊗ |1〉j = w/u. There are 22 × 22 = 16 matrix elements, and 6 ofthem [Rij(u,w)]

δ1δ2ǫ1ǫ2 satisfying ǫ1 + ǫ2 = δ1 + δ2 are displayed. The other matrix elements are

identically zero.

which acts onWi⊗Wj⊗Wk. Note that Rij(u, v), Rik(u,w) and Rjk(v,w) are operators actingon Wi⊗Wj ⊗Wk, and acts nontrivially on Wi⊗Wj, Wi⊗Wk and Wj ⊗Wk respectively, andacts as identity on the remaining space. In terms of matrix entries, the Yang-Baxter relation(2.11) means that

∑

γ1,γ2,γ3=0,1

[R(u, v)]δ1δ2γ3γ1 [R(u,w)]γ3δ3ǫ1γ2 [R(v,w)]γ1γ2ǫ2ǫ3

=∑

γ1,γ2,γ3=0,1

[R(v,w)]δ2δ3γ2γ3 [R(u,w)]δ1γ3γ1ǫ3 [R(u, v)]γ1γ2ǫ1ǫ2 , (2.12)

holds for ǫ1, ǫ2, ǫ3, δ1, δ2, δ3 = 0, 1. Acting both hand sides of (2.11) on |ǫ1〉i ⊗ |ǫ2〉j ⊗ |ǫ3〉kand taking coefficients of |δ1〉i ⊗ |δ2〉j ⊗ |δ3〉k give (2.12). Note that Rij(u, v), Rik(u,w) andRjk(v,w) act on Wi ⊗Wj ⊗Wk, hence are 23 × 23 matrices, and we used relations (2.10) toexpress the matrix elements in terms of [R(u, v)]γ1γ2ǫ1ǫ2 etc. For example, [R(u, v)]γ1γ2ǫ1ǫ2 in theright hand side of (2.12) comes from the following computation: Rij(u, v)|ǫ1〉i⊗|ǫ2〉j⊗|ǫ3〉k =∑

γ1,γ2,γ=0,1

[Rij(u, v)]γ1γ2γǫ1ǫ2ǫ3 |γ1〉i ⊗ |γ2〉j ⊗ |γ〉k =

∑

γ1,γ2,γ=0,1

[R(u, v)]γ1γ2ǫ1ǫ2 δǫ3γ |γ1〉i ⊗ |γ2〉j ⊗ |γ〉k =

6

Figure 2: The Yang-Baxter relation (2.11). The left and right figure representsRij(u, v)Rik(u,w)Rjk(v,w) and Rjk(v,w)Rik(u,w)Rij(u, v) respectively.

∑

γ1,γ2=0,1

[R(u, v)]γ1γ2ǫ1ǫ2 |γ1〉i ⊗ |γ2〉j ⊗ |ǫ3〉k.

We remark that more generally, (2.11) can be viewed as a relation in W1 ⊗ · · · ⊗Wm foran integer m ≥ 3 where Rij(u, v) acts nontrivially on Wi and Wj, Rik(u,w) on Wi and Wk,Rjk(v,w) on Wj and Wk, and acts as identity on the remaining space. The relation (2.11)also holds as operators acting on W1 ⊗ · · · ⊗Wm due to the identities (2.12).

We next construct the monodromy matrix Ta(u)

Ta(u) = Ra,m(u, 1) · · ·Ra1(u, 1), (2.13)

which acts on Wa ⊗W1 ⊗ · · · ⊗Wm. The operator Ra,j(u, 1) (1 ≤ j ≤ m) acts on Wa andWj nontrivially and as identity on the other spaces, and the space Wa which every R-matrixused to construct the monodromy matrix acts nontrivially is called as the auxiliary space.To distinguish the auxiliary space from the other spaces W1, . . . ,Wm, we use alphabets, a forexample, to label this space and denote the vector space as Wa.

The monodromy matrix Ta(u) can be regarded as a 2m+1 × 2m+1 matrix since it acts onthe 2m+1-dimensional space Wa ⊗W1 ⊗ · · · ⊗Wm. The matrix elements of the monodromymatrix [T (u)]δa,δ1,...,δmǫa,ǫ1,...,ǫm for ǫa, ǫ1, . . . , ǫm, δa, δ1, . . . , δm = 0, 1 defined as

Ta(u)|ǫa〉a ⊗ |ǫ1〉1 ⊗ · · · ⊗ |ǫm〉m =∑

δa,δ1,...,δm=0,1

[T (u)]δa,δ1,...,δmǫa,ǫ1,...,ǫm |δa〉a ⊗ |δ1〉1 ⊗ · · · ⊗ |δm〉m,

(2.14)

7

can be written using the R-matrix elements as

[T (u)]δa,δ1,...,δmǫa,ǫ1,...,ǫm =∑

γ1,...,γm−1=0,1

[R(u, 1)]δaδmγm−1ǫm

m−2∏

j=1

[R(u, 1)]γj+1δj+1γjǫj+1 [R(u, 1)]γ1δ1ǫaǫ1 . (2.15)

The matrix elements of the monodromy matrix Ta(u) with respect to the auxiliary spaceWa are called as the A- B- C- and D-operators. Namely, we write the results of the actionof the operator Ta(u) on the basis vectors |0〉a, |1〉a of the space Wa as

Ta(u)|0〉a = |0〉a ⊗A(u) + |1〉a ⊗ C(u), (2.16)

Ta(u)|1〉a = |0〉a ⊗B(u) + |1〉a ⊗D(u), (2.17)

and the coefficients A(u), B(u), C(u) and D(u) are the ABCD-operators. Note that A(u),B(u), C(u) and D(u) act on the 2m-dimensional vector space W1 ⊗ · · · ⊗ Wm, and can be

regarded as 2m×2m matrices. By definition (2.16), (2.17), the matrix elements [A(u)]δ1,...,δmǫ1,...,ǫm ,

[B(u)]δ1,...,δmǫ1,...,ǫm , [C(u)]δ1,...,δmǫ1,...,ǫm , [D(u)]δ1,...,δmǫ1,...,ǫm for ǫ1, . . . , ǫm, δ1, . . . , δm = 0, 1 defined as

A(u)|ǫ1〉1 ⊗ · · · ⊗ |ǫm〉m =∑

δ1,...,δm=0,1

[A(u)]δ1,...,δmǫ1,...,ǫm |δ1〉1 ⊗ · · · ⊗ |δm〉m, (2.18)

B(u)|ǫ1〉1 ⊗ · · · ⊗ |ǫm〉m =∑

δ1,...,δm=0,1

[B(u)]δ1,...,δmǫ1,...,ǫm |δ1〉1 ⊗ · · · ⊗ |δm〉m, (2.19)

C(u)|ǫ1〉1 ⊗ · · · ⊗ |ǫm〉m =∑

δ1,...,δm=0,1

[C(u)]δ1,...,δmǫ1,...,ǫm |δ1〉1 ⊗ · · · ⊗ |δm〉m, (2.20)

D(u)|ǫ1〉1 ⊗ · · · ⊗ |ǫm〉m =∑

δ1,...,δm=0,1

[D(u)]δ1,...,δmǫ1,...,ǫm |δ1〉1 ⊗ · · · ⊗ |δm〉m, (2.21)

can be obtained from the monodromy matrix elements [T (u)]δa,δ1,...,δmǫa,ǫ1,...,ǫm by specializing ǫa andδa as follows:

[A(u)]δ1,...,δmǫ1,...,ǫm = [T (u)]0,δ1,...,δm0,ǫ1,...,ǫm, [B(u)]δ1,...,δmǫ1,...,ǫm = [T (u)]0,δ1,...,δm1,ǫ1,...,ǫm

, (2.22)

[C(u)]δ1,...,δmǫ1,...,ǫm = [T (u)]1,δ1,...,δm0,ǫ1,...,ǫm, [D(u)]δ1,...,δmǫ1,...,ǫm = [T (u)]1,δ1,...,δm1,ǫ1,...,ǫm

. (2.23)

Using the bra-ket notation, the ABCD-operators are written as

A(u) = a〈0|Ta(u)|0〉a, (2.24)

B(u) = a〈0|Ta(u)|1〉a, (2.25)

C(u) = a〈1|Ta(u)|0〉a, (2.26)

D(u) = a〈1|Ta(u)|1〉a. (2.27)

In this paper, we focus on commutation relations and partition functions constructedfrom the B-operator (2.25) and the D-operator (2.27) (Figure 3).

Let us review the commutation relations in this section. The ABCD-operators satisfycommutation relations which can be obtained from the intertwining relation

Rab(u1, u2)Ta(u1)Tb(u2) = Tb(u2)Ta(u1)Rab(u1, u2), (2.28)

8

Figure 3: The B-operator (2.25) (top) and the D-operator (2.27) (bottom).

acting on Wa ⊗ Wb ⊗ W1 ⊗ · · · ⊗ Wm. Here, Rab(u1, u2) acts nontrivially on Wa ⊗ Wb

and as identity on W1 ⊗ · · · ⊗ Wm. Ta(u1) and Tb(u2) are monodromy matrices acting onWa ⊗Wb ⊗W1 ⊗ · · · ⊗Wm which naturally extend the monodromy matrix (2.13). Ta(u1) :=Ra,m(u1, 1) · · ·Ra1(u1, 1) acts on Wa,W1, . . . ,Wm nontrivially and as identity on Wb, and

Tb(u2) := Rb,m(u2, 1) · · ·Rb1(u2, 1) acts on Wb,W1, . . . ,Wm nontrivially and as identity on

Wa. Note that Ta(u1) can be regarded as a 2m+2 × 2m+2 matrix, and since it acts on Wb

trivially, the matrix elements of Ta(u1) with respect to the space Wb are given as

b〈0|Ta(u1)|0〉b = b〈1|Ta(u1)|1〉b = Ta(u1), b〈0|Ta(u1)|1〉b = b〈1|Ta(u1)|0〉b = 0, (2.29)

using the bra-ket notation. Here, Ta(u1) is the monodromy matrix (2.13) which acts onWa ⊗W1 ⊗ · · · ⊗Wm which we introduced before, and 0 is the 2m+1 × 2m+1 zero matrix.

Similarly, we note that Tb(u2) can be regarded as a 2m+2 × 2m+2 matrix which acts onWa trivially, which means that the matrix elements with respect to the space Wa are givenas

a〈0|Tb(u2)|0〉a = a〈1|Tb(u2)|1〉a = Tb(u2), a〈0|Tb(u2)|1〉a = a〈1|Tb(u2)|0〉a = 0, (2.30)

where Tb(u2) = Rb,m(u2, 1) · · ·Rb1(u2, 1) is the monodromy matrix which acts on Wb ⊗W1 ⊗· · · ⊗Wm.

The intertwining relation (2.28) follows by applying the Yang-Baxter relation (2.11) re-

9

peatedly. For example, the m = 2 case of (2.28) can be shown as follows:

Rab(u1, u2)Ta(u1)Tb(u2)

=Rab(u1, u2)Ra2(u1, 1)Ra1(u1, 1)Rb2(u2, 1)Rb1(u2, 1)

=Rab(u1, u2)Ra2(u1, 1)Rb2(u2, 1)Ra1(u1, 1)Rb1(u2, 1)

=Rb2(u2, 1)Ra2(u1, 1)Rab(u1, u2)Ra1(u1, 1)Rb1(u2, 1)

=Rb2(u2, 1)Ra2(u1, 1)Rb1(u2, 1)Ra1(u1, 1)Rab(u1, u2)

=Rb2(u2, 1)Rb1(u2, 1)Ra2(u1, 1)Ra1(u1, 1)Rab(u1, u2)

=Tb(u2)Ta(u2)Rab(u1, u2). (2.31)

In the second equality in (2.31), we used Ra1(u1, 1)Rb2(u2, 1) = Rb2(u2, 1)Ra1(u1, 1). Thiscommutativity holds since Ra1(u1, 1) acts nontrivially only onWa andW1 and acts as identityon the other spaces, in particular Wb and W2, and Rb2(u2, 1) acts nontrivially only on Wb

and W2 and acts as identity on the rest, in particular Wa and W1, hence the result of actingRa1(u1, 1) and Rb2(u2, 1) successively on any vector in Wa ⊗Wb ⊗W1 ⊗ · · · ⊗Wm gives thesame result and is independent of the order of the action of the two operators. We can also seethat Ra2(u1, 1)Rb1(u2, 1) = Rb1(u2, 1)Ra2(u1, 1) holds by the same arugment which is used inthe fifth equality. In the third and fourth equalities, we used the Yang-Baxter relation (2.11).

Acting both sides of (2.28) on |ǫa〉a ⊗ |ǫb〉b ∈ Wa ⊗Wb and noting Ta(u1) and Tb(u2) actstrivially on Wb and Wa respectively, we get

∑

δa,δb,γa,γb=0,1

|γa〉a ⊗ |γb〉b [R(u1, u2)]γaγbδaδb

× a〈δa| ⊗ b〈δb|Ta(u1)|ǫa〉a ⊗ |δb〉b a〈ǫa| ⊗ b〈δb|Tb(u2)|ǫa〉a ⊗ |ǫb〉b

=∑

δa,δb,γa,γb=0,1

|γa〉a ⊗ |γb〉b a〈γa| ⊗ b〈γb|Tb(u2)|γa〉a ⊗ |δb〉b

× a〈γa| ⊗ b〈δb|Ta(u1)|δa〉a ⊗ |δb〉b [R(u1, u2)]δaδbǫaǫb

. (2.32)

From (2.29) and (2.30), we have

a〈δa| ⊗ b〈δb|Ta(u1)|ǫa〉a ⊗ |δb〉b = a〈δa|Ta(u1)|ǫa〉a, (2.33)

a〈ǫa| ⊗ b〈δb|Tb(u2)|ǫa〉a ⊗ |ǫb〉b = b〈δb|Tb(u2)|ǫb〉b, (2.34)

a〈γa| ⊗ b〈γb|Tb(u2)|γa〉a ⊗ |δb〉b = b〈γb|Tb(u2)|δb〉b, (2.35)

a〈γa| ⊗ b〈δb|Ta(u1)|δa〉a ⊗ |δb〉b = a〈γa|Ta(u1)|δa〉a, (2.36)

and inserting these into (2.32), we get the following commutation relations between theABCD-operators

∑

δa,δb=0,1

[R(u1, u2)]γaγbδaδb a〈δa|Ta(u1)|ǫa〉a b〈δb|Tb(u2)|ǫb〉b

=∑

δa,δb=0,1

b〈γb|Tb(u2)|δb〉b a〈γa|Ta(u1)|δa〉a [R(u1, u2)]δaδbǫaǫb

. (2.37)

For example, the ǫa = ǫb = γa = 1, γb = 0 case of (2.37) is

(1− u2/u1)D(u1)B(u2) + u2/u1B(u1)D(u2) = B(u2)D(u1). (2.38)

10

We rewrite (2.38) as

D(u1)B(u2) =1

1− u2/u1B(u2)D(u1) +

1

1− u1/u2B(u1)D(u2), (2.39)

for later purpose. Some other cases of the commutation relations (2.37) which are used inthis paper are

D(u1)B(u2) = D(u2)B(u1), (2.40)

B(u1)B(u2) = B(u2)B(u1), (2.41)

D(u1)D(u2) = D(u2)D(u1). (2.42)

Note that (2.39), (2.40), (2.41) and (2.42) are relations which act on W1 ⊗ · · · ⊗ Wm sinceB(u) and D(u) are operators acting on W1 ⊗ · · · ⊗Wm.

Now recall the following notations for the sets given in the introduction. Snk is a k-subset

of [n] = {1, 2, . . . , n}, the set of k-subsets of n is denoted as([n]k

), and Sn

k = {1, 2, . . . , n}\Snk .

The following commutation relation between the multiple operators∏nj=k+1D(uj) and

∏kj=1B(uj)

n∏

j=k+1

D(uj)

k∏

j=1

B(uj) =∑

Snk∈([n]

k )

∏

i∈Snk,j∈Sn

k

1

1− ui/uj

∏

i∈Snk

B(ui)∏

j∈Snk

D(uj), (2.43)

is one of the commutation relations between multiple operators essentially derived by Shigechiand Uchiyama [35] in their study of integrable models whose L-operators satisify the RLLYang-Baxter relation with (a gauge transformed version of) the five-vertex model R-matrix(2.5) intertwining the L-operators. Various multiple commutation relations including thistype were investigated in [35] for the purpose of studying generalized scalar products. Onecan also use this commutation relation to give another proof [80] of an identity for the factorialGrothendieck polynomials [75], which is an extension of the one for the Schur polynomials[76].

We record the argument given in [35] (which appears in Proof of Theorem 6.1) for com-pleteness. First, using (2.39), (2.41) and (2.42) to move all the B-operators to the left of allthe D-operators, one notes the operator part of all the terms which appear can be expressedas

∏

i∈Snk

B(ui)∏

j∈Snk

D(uj), (2.44)

for Snk ∈

([n]

k

). To extract the coefficient of (2.44) for a fixed Sn

k , one uses (2.40) repeatedly

to rewrite the left hand side of (2.43) as

∏

j∈Snk

D(uj)∏

i∈Snk

B(ui). (2.45)

Finally, we only use (2.39) repeatedly to move all the B-operators to the left of all the D-operators. We only need to concentrate on the first term of the right hand side of (2.39) when

11

commuting the B- and D-operators to extract the coefficient of (2.44), since if we once usethe second term of the left hand side of (2.39), we get other operators. Noting this, one finds

the coefficient of the operator is given by∏

i∈Snk,j∈Sn

k

1

1− ui/uj, and hence the commutation

relation (2.43) follows.

3 Partition functions

In this section, we introduce several types of partition functions which are used in this paper,and give their explicit forms using one version of skew Grothendieck polynomials.

Figure 4: The partition functions Zm,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+n) constructed fromthe B-operators (3.3). The figure represents the case m = 11, n = 5, ℓ = 3, (x1, x2, x3) =(2, 5, 7), (y1, y2, y3, y4, y5, y6, y7, y8) = (2, 3, 4, 6, 7, 8, 9, 11).

For an increasing sequence of integers x = (x1, x2, . . . , xℓ) and an integer j, let us defineδj,x as δj,x = 1 if there exists xk (1 ≤ k ≤ ℓ) such that j = xk, and δj,x = 0 otherwise. Foreach increasing sequence of integers 1 ≤ x1 < x2 < · · · < xℓ ≤ m, we define the configurationvector and its dual as

|x1 · · · xℓ〉m := ⊗mj=1(|0〉j + δj,x(|1〉j − |0〉j)) ∈ W1 ⊗ · · · ⊗Wm, (3.1)

and

m〈x1 · · · xℓ| := ⊗mj=1(j〈0| + δj,x(j〈1| − j〈0|)) ∈ W ∗

1 ⊗ · · · ⊗W ∗m. (3.2)

For example, |3, 5〉6 = |0〉1 ⊗ |0〉2 ⊗ |1〉3 ⊗ |0〉4 ⊗ |1〉5 ⊗ |0〉6.By using the B-operators, we introduce the first type of partition functions

Zm,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+n) for two increasing sequences of integers 1 ≤ x1 <x2 < · · · < xℓ ≤ m and 1 ≤ y1 < y2 < · · · < yℓ+n ≤ m as (Figure 4)

Zm,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+n) = m〈y1 · · · yℓ+n|B(u1) · · ·B(un)|x1 · · · xℓ〉m. (3.3)

12

Figure 5: The matrix element m〈y1 · · · yℓ+1|B(u)|x1 · · · xℓ〉m which we takeas a definition of Gλ//µ(1 − u−1). The figure represents the case m = 9,ℓ = 3, (x1, x2, x3) = (3, 5, 7), (y1, y2, y3, y4) = (2, 5, 6, 9). The matrix el-ement of the B-operator can be computed as 9〈2, 5, 6, 9|B(u)|3, 5, 7〉9 =[R(u, 1)]0110[R(u, 1)]1010[R(u, 1)]1001[R(u, 1)]0110[R(u, 1)]1111[R(u, 1)]1010[R(u, 1)]1001[R(u, 1)]0110[R(u, 1)]1010 =1× (1−u−1)×u−1×1×1× (1−u−1)×u−1×1× (1−u−1) = (1−u−1)3u−2. We can computegraphically by multiplying all nine R-matrix elements 1 − u−1, 1, u−1, (1 − u−1), 1, 1, u−1,1 − u−1, 1 in the figure (see also Figure 1). In terms of skew Grothendieck polynomials, λand µ in this example are λ = (λ1, λ2, λ3, λ4) = (y4− 4, y3− 3, y2− 2, y3− 1) = (5, 3, 3, 1) andµ = (µ1, µ2, µ3) = (x3 − 3, x2− 2, x1 − 1) = (4, 3, 2), so this matrix element of the B-operatorcorresponds to G(5,3,3,1)//(4,3,2)(1 − u−1). One can check the right hand side of (3.8) whenλ = (5, 3, 3, 1), µ = (4, 3, 2) is (1− u−1)3u−2.

For ℓ = 0, we define Zm,n(u1, . . . , un|φ|y1, . . . , yn) as

Zm,n(u1, . . . , un|φ|y1, . . . , yn) := m〈y1 · · · yn|B(u1) · · ·B(un)|0〉1 ⊗ · · · ⊗ |0〉m. (3.4)

The partition functions Zm,n(u1, . . . , un|φ|y1, . . . , yn) are called as the off-shell Bethe wave-functions of the five-vertex model, and can be expressed using the Grassmannian Grothendieckpolynomials [15, 17, 18, 19, 81], whose determinant representation is given by [18, 21, 82, 83]

Gλ(z1, . . . , zn) =det(z

λj+n−ji (1− zi)

j−1)i≤i,j≤n∏1≤i<j≤n(zi − zj)

, (3.5)

where {z1, . . . , zn} is a set of commuting variables and λ = (λ1, λ2, . . . , λn) is a partition, i.e.a nonincreasing sequence of nonnegative integers whose graphical representation is naturallygiven by the Young diagram.

The off-shell Bethe wavefunctions (3.4) of the integrable five-vertex model is a partitionfunction representation of the Grothendieck polynomials [4, 5, 9, 10].

Proposition 3.1. ([4] Lemma 5.2, [5] Thm 2.2, [9] Prop 4.1, [10] Prop 1) We have

Zm,n(u1, . . . , un|φ|y1, . . . , yn) = Gλ(1− u−11 , . . . , 1− u−1

n ), (3.6)

13

where λ and y1, . . . , yn are related by λj = yn−j+1 − n+ j − 1 (j = 1, . . . , n).

In view of the correspondence (3.6), we introduce a version of the skew Grothendieckpolynomials with a single variable as the matrix elements of a single B-operator:

Gλ//µ(1− u−1) = m〈y1 · · · yℓ+1|B(u)|x1 · · · xℓ〉m, (3.7)

where λj = yℓ−j+2 − ℓ+ j − 2 (j = 1, . . . , ℓ+ 1) and µj = xℓ−j+1 − ℓ+ j − 1 (j = 1, . . . , ℓ).One can compute the matrix elements of a single B-operator m〈y1 · · · yℓ+1|B(u)|x1 · · · xℓ〉m

explicitly (see [5] Prop 3.4 for example), and through the identification (3.7), the skewGrothendieck polynomials Gλ//µ(1− u−1) is explicitly given by

Gλ//µ(1− u−1) =

{(1− u−1)

∑ℓ+1j=1 λj−

∑ℓj=1 µj

∏ℓj=1{u

−1 + δλj+1,µj(1− u−1)}, λ ≻ µ,

0, otherwise.

(3.8)

Here, λ ≻ µ is defined as the following relation between λ and µ

λ1 ≥ µ1 ≥ λ2 ≥ µ2 ≥ · · · ≥ µℓ ≥ λℓ+1, (3.9)

which is called as the interlacing relation. See Figure 5 for an example of (3.8), and [5] Prop3.4 for a proof of (3.8) (the argument in the proof of Prop 3.4 applied to the R-matrix (2.5)gives (3.8). The R-matrix in [5] and (2.5) are related by gauge transformation).

Let us give a remark. We used the notation Gλ//µ(1−u−1) here instead of Gλ/µ(1−u−1).There are two versions of skew Grothendieck polynomials Gλ/µ(x) and Gλ//µ(x) introducedby Buch in [18]. Gλ/µ(x) is defined as certain types of stable Grothendieck polynomialswhich are originally labeled by permutations, and Gλ//µ(x) is introduced as a natural skewversion of the stable Grothendieck polynomials from the coproduct structure. A furtherstudy of this type of skew Grothendieck polynomials was done by Yeliussizov in [84], inwhich skew Grothendieck polynomials with finite number of variables is first defined usingSchur operators, and skew Grothendieck polynomials with infinitely many variables is definedas a limit. We note that the explicit expression for one variable case (3.8) which we definedas matrix elements of the B-operators is the same with the β = 1 case of the one given inProposition 4.3 in [84], hence we use this notation. The skew Grothendieck polynomials withmultivariables which we later define as (3.18) can also be shown to be the same with the onewhich is introduced using Schur operators in [84], Definition 4.1. See Appendix A for details.

We can show the following relation

Gλ(1− u−11 , . . . , 1 − u−1

ℓ , 1− u−1ℓ+1) =

∑

λ≻µ

Gλ//µ(1− u−1ℓ+1)Gµ(1− u−1

1 , . . . , 1− u−1ℓ ), (3.10)

14

using the lattice model construction as follows:

Gλ(1− u−11 , . . . , 1− u−1

ℓ , 1− u−1ℓ+1)

=m〈y1 · · · yℓ+1|B(u1) · · ·B(uℓ+1)|0〉1 ⊗ · · · ⊗ |0〉m

=m〈y1 · · · yℓ+1|B(uℓ+1)× Id×B(u1) · · ·B(uℓ)|0〉1 ⊗ · · · ⊗ |0〉m

=m〈y1 · · · yℓ+1|B(uℓ+1)×

(|0〉1 ⊗ · · · ⊗ |0〉m 1〈0| ⊗ · · · ⊗ m〈0|

+

m∑

j=1

∑

1≤x1<···<xj≤m

|x1 · · · xj〉m m〈x1 · · · xj |

)×B(u1) · · ·B(uℓ)|0〉1 ⊗ · · · ⊗ |0〉m

=∑

1≤x1<···<xℓ≤m

m〈y1 · · · yℓ+1|B(uℓ+1)|x1 · · · xℓ〉m m〈x1 · · · xℓ|B(u1) · · ·B(uℓ)|0〉1 ⊗ · · · ⊗ |0〉m

=∑

λ≻µ

Gλ//µ(1− u−1ℓ+1)Gµ(1− u−1

1 , . . . , 1− u−1ℓ ). (3.11)

Here we used the commutativity of B-operators (2.41) in the second equality. We also insertedthe identity operator on W1 ⊗ · · · ⊗Wm

Id = |0〉1 ⊗ · · · ⊗ |0〉m 1〈0| ⊗ · · · ⊗ m〈0|+m∑

j=1

∑

1≤x1<···<xj≤m

|x1 · · · xj〉m m〈x1 · · · xj|, (3.12)

which is one way of rewriting the standard expression for the identity operator

Id =∑

ǫ1,ǫ2,...,ǫm=0,1

|ǫ1〉1 ⊗ |ǫ2〉2 ⊗ · · · ⊗ |ǫm〉m 1〈ǫ1| ⊗ 2〈ǫ2| ⊗ · · · ⊗ m〈ǫm|. (3.13)

The expression above follows from the bra-ket expression for the identity operator |0〉〈0| +

|1〉〈1| =∑

ǫ=0,1

|ǫ〉〈ǫ| on the two-dimensional vector space W .

From the ice-rule for the R-matrix, we note that B(u1) · · ·B(uℓ)|0〉1 ⊗ · · · ⊗ |0〉m is writ-ten as a linear combination of |x1 · · · xℓ〉m, 1 ≤ x1 < · · · < xℓ ≤ m, which means that

m〈x1 · · · xj |B(u1) · · ·B(uℓ)|0〉1 ⊗ · · · ⊗ |0〉m = 0 unless j = ℓ. Hence, we can restrict theinserted identity operator to

∑

1≤x1<···<xℓ≤m

|x1 · · · xℓ〉m m〈x1 · · · xℓ|, (3.14)

which we used in the fourth equality. Let us call the argument of restricting the insertedidentity operator (3.12) to (3.14) as the restriction argument.

The relation (3.10) can be regarded as a K-theoretic version of the following relation forthe Schur and skew Schur functions

sλ(x1, . . . , xℓ, xℓ+1) =∑

λ≻µ

sλ/µ(xℓ+1)sµ(x1, . . . , xℓ), (3.15)

and as a finite variable truncation of the definition of the skew stable Grothendieck polyno-mials using coproduct ∆ : Γ → Γ⊗ Γ with (∆f)(x, y) = f(x, y) (Buch [18], (6.3))

∆(Gλ) =∑

µ⊂λ

Gµ ⊗Gλ//µ. (3.16)

15

Here, Γ = ⊕λZ ·Gλ where Gλ are stable Grothendieck polynomials.From (3.16), we have ([18], section 9, [84], Remark 4.2)

Gλ//λ(1− u−11 , 1− u−1

2 , . . . ) =∏

k≥1

u−i(λ)k , (3.17)

where i(λ) is the number of indices i such that λi > λi+1. This is different from Gλ/λ =

Gφ = 1 in general. Truncating to the one variable case gives Gλ//λ(1− u−11 ) = u

−i(λ)1 , which

matches the definition of the skew Grothendieck polynomials for one variable Gλ//(1−u−1) as

matrix elements of the B-operator (3.7). For example, G(1)//(1)(1−u−11 ) = 3〈1, 3|B(u1)|2〉3 =

[R(u1, 1)]0110[R(u1, 1)]

1001[R(u1, 1)]

0110 = u−1

1 = u−i((1))1 .

We introduce the skew Grothendieck polynomials with multi variables by composing skewGrothendieck polynomials of a single variable as

Gλ//µ(1− u−11 , . . . , 1− u−1

n ) =∑

λ=λ(0)≻λ(1)≻···≻λ(n)=µ

n∏

j=1

Gλ(j−1)//λ(j)(1− u−1j ). (3.18)

The following determinant formula for the skew Grothendieck polynomials is shown byIwao [14] by the boson-fermion correspondence.

Proposition 3.2. (Iwao [14], β = −1 case of Proposition 4.8) For n ≤ r − ℓ(µ), we have

Gλ//µ(z1, . . . , zn) = det

( µj−j+r∑

k=0

(1− j

k

)H

(i−1)λi−µj−i+j+k(z1, . . . , zn)

)

1≤i,j≤r

, (3.19)

where

H(i)p (z1, . . . , zn) :=

∞∑

ℓ=0

(i

ℓ

)(−1)ℓhp+ℓ(z1, . . . , zn). (3.20)

Here, binomials are defined as

(p

q

)=

∏q−1j=0(p − j)

q!for p, q integers, and hi(z1, . . . , zn) is the

i-th complete symmetric polynomial with commuting variables z1, . . . , zn.

The partition functions Zm,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+n) (3.3) are constructed bytaking matrix elements of multiple B-operators. By inserting the identity operator (3.12)between the B-operators and using the restriction argument and (3.7), the partition functionscan be rewritten as the right hand side of (3.18), hence the following correspondence holds:

Zm,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+n) = Gλ//µ(1− u−11 , . . . , 1− u−1

n ), (3.21)

where λj = yℓ−j+n+1−ℓ+j−n−1 (j = 1, . . . , ℓ+n) and µj = xℓ−j+1−ℓ+j−1 (j = 1, . . . , ℓ).Let us illustrate the n = 2 case of showing (3.21) (the general case can be shown in the

16

Figure 6: The partition functions Um,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ) constructed fom theD-operators (3.23). The figure represents the case m = 11, n = 5, ℓ = 4, (x1, x2, x3, x4) =(2, 5, 7, 9), (y1, y2, y3, y4) = (3, 6, 9, 11).

same way). This can be shown in the following way

Zm,2(u1, u2|x1, . . . , xℓ|y1, . . . , yℓ+2)

=m〈y1 · · · yℓ+2|B(u1)B(u2)|x1 · · · xℓ〉m

=m〈y1 · · · yℓ+2|B(u1)× Id×B(u2)|x1 · · · xℓ〉m

=m〈y1 · · · yℓ+2|B(u1)×

(|0〉1 ⊗ · · · ⊗ |0〉m 1〈0| ⊗ · · · ⊗ m〈0|

+

m∑

j=1

∑

1≤z1<···<zj≤m

|z1 · · · zj〉m m〈z1 · · · zj |

)×B(u2)|x1 · · · xℓ〉m

=∑

1≤z1<···<zℓ+1≤m

m〈y1 · · · yℓ+2|B(u1)|z1 · · · zℓ+1〉m m〈z1 · · · zℓ+1|B(u2)|x1 · · · xℓ〉m

=∑

λ≻λ(1)≻µ

Gλ//λ(1) (1− u−11 )Gλ(1)//µ(1− u−1

2 )

=Gλ//µ(1− u−11 , 1− u−1

2 ), (3.22)

where λ(1) = (λ(1)1 , . . . , λ

(1)ℓ+1) is defined as λ

(1)j = zℓ−j+2 − ℓ+ j − 2 (j = 1, . . . , ℓ+ 1) . Note

that we used the property m〈z1 · · · zj |B(u2)|x1 · · · xℓ〉m = 0 unless j = ℓ + 1 in the fourthequality in (3.22) which comes from the ice-rule of the R-matrix, and used (3.7) and (3.8) inthe fifth equality.

Next, we introduce the second type of partition functions constructed from theD-operators.Define Um,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ) for two increasing sequences of integers 1 ≤ x1 <x2 < · · · < xℓ ≤ m and 1 ≤ y1 < y2 < · · · < yℓ ≤ m as (Figure 6)

Um,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ) = m〈y1 · · · yℓ|D(u1) · · ·D(un)|x1 · · · xℓ〉m. (3.23)

17

Figure 7: The partition functions Zm+n,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ,m + 1, . . . ,m + n)(3.24). The figure represents the case m = 11, n = 5, ℓ = 4, (x1, x2, x3, x4) = (2, 5, 7, 9),(y1, y2, y3, y4) = (3, 6, 9, 11).

The partition functions (3.23) introduced as matrix elements of multiple D-operators, canbe connected with partition functions of the first type which are constructed from the B-operators, and hence with the skew Grothendieck polynomials. Consider the following parti-tion functions constructed from the B-operators (Figure 7)

Zm+n,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ,m+ 1, . . . ,m+ n)

=m+n〈y1 · · · yℓ,m+ 1, . . . ,m+ n|B(m+n)(u1) · · ·B(m+n)(un)|x1 · · · xℓ〉m+n, (3.24)

where

B(m+n)(u) = a〈0|Ra,m+n(u) · · ·Ra1(u)|1〉a ∈ End(W1 ⊗ · · · ⊗Wm+n). (3.25)

We can show that the matrix elements of multiple D-operators (3.23) are the same withthose of multiple B-operators (3.24).

Lemma 3.3. The following relation holds:


=Um,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ). (3.26)

In Appendix B, a full proof of (3.26) using equations only is given. We also give agraphical derivation of (3.26) in Appendix C.

Applying the correspondence between partition functions and skew Grothendieck polyno-mials (3.21) to Zm+n,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ,m+1, . . . ,m+n) and combining with(3.26), one finds that the partition functions of the D-operatorsUm,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ) are explicitly given by the skew Grothendieck polyno-mials as

Um,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ) = G((m−ℓ)n ,λ)//µ(1− u−11 , . . . , 1− u−1

n ), (3.27)

18

Figure 8: The mixed partition functions ZBDm,n,k(u1, . . . , uk|uk+1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+k)constructed from both the B- and D-operators (3.28). The figure represents the case m = 11,n = 5, ℓ = 4, k = 2 (x1, x2, x3, x4) = (2, 4, 5, 7), (y1, y2, y3, y4, y5, y6) = (2, 4, 7, 8, 9, 11).

where λj = yℓ−j+1 − ℓ+ j − 1 (j = 1, . . . , ℓ) and µj = xℓ−j+1 − ℓ+ j − 1 (j = 1, . . . , ℓ).In this section, we also introduce two types of partition functions which are constructed

from both the B-operators and the D-operators, and describe their explicit forms using skewGrothendieck polynoimals.

The first type of mixed partition functions is (Figure 8)

ZBDm,n,k(u1, . . . , uk|uk+1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+k)

=m〈y1 · · · yℓ+k|B(u1) · · ·B(uk)D(uk+1) · · ·D(un)|x1 · · · xℓ〉m. (3.28)

By inserting the identity operator (3.12) between B(uk) andD(uk+1) and using the restrictionargument and the correspondences (3.21) and (3.27), (3.28) can be described as a sum ofproducts of the skew Grothendieck polynomials as


=∑

ν⊆(m−ℓ)ℓ

Gλ//ν(1− u−11 , . . . , 1− u−1

k )G((m−ℓ)n−k ,ν)//µ(1− u−1k+1, . . . , 1− u−1

n ), (3.29)

where λj = yℓ−j+k+1−ℓ+j−k−1 (j = 1, . . . , ℓ+k) and µj = xℓ−j+1−ℓ+j−1 (j = 1, . . . , ℓ).

19

(3.29) can be shown as follows


=m〈y1 · · · yℓ+k|B(u1) · · ·B(uk)D(uk+1) · · ·D(un)|x1 · · · xℓ〉m

=m〈y1 · · · yℓ+k|B(u1) · · ·B(uk)× Id×D(uk+1) · · ·D(un)|x1 · · · xℓ〉m

=m〈y1 · · · yℓ+k|B(u1) · · ·B(uk)×

(|0〉1 ⊗ · · · ⊗ |0〉m 1〈0| ⊗ · · · ⊗ m〈0|

+

m∑

j=1

∑

1≤z1<···<zj≤m

|z1 · · · zj〉m m〈z1 · · · zj |

)×D(uk+1) · · ·D(un)|x1 · · · xℓ〉m

=∑

1≤z1<···<zℓ≤m

m〈y1 · · · yℓ+k|B(u1) · · ·B(uk)|z1 · · · zℓ〉m

× m〈z1 · · · zℓ|D(uk+1) · · ·D(un)|x1 · · · xℓ〉m

=∑

1≤z1<···<zℓ≤m

Zm,k(u1, . . . , uk|z1, . . . , zℓ|y1, . . . , yℓ+k)

× Um,n−k(uk+1, . . . , un|x1, . . . , xℓ|z1, . . . , zℓ)

=∑

ν⊆(m−ℓ)ℓ

Gλ//ν(1− u−11 , . . . , 1− u−1

k )G((m−ℓ)n−k ,ν)//µ(1− u−1k+1, . . . , 1− u−1

n ), (3.30)

where ν = (ν1, . . . , νℓ) is defined as νj = zℓ−j+1 − ℓ + j − 1 (j = 1, . . . , ℓ). Note that weused m〈y1 · · · yℓ+k|B(u1) · · ·B(uk)|z1 · · · zj〉m = 0 unless j = ℓ in the fourth equality in (3.30)which follows from the ice-rule of the R-matrix, and (3.21), (3.27) in the last equality.

Figure 9: The mixed partition functions ZDBm,n,k(uk+1, . . . , un|u1, . . . , uk|x1, . . . , xℓ|y1, . . . , yℓ+k)constructed from both the B- and D-operators (3.31). Note the order of the layersof the B-operators and those of the D-operators is reversed from (3.28). The fig-ure represents the case m = 11, n = 5, ℓ = 4, k = 2 (x1, x2, x3, x4) = (2, 4, 5, 7),(y1, y2, y3, y4, y5, y6) = (2, 4, 7, 8, 9, 11).

20

Figure 10: The partition functions Zm+n−k,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+k,m+1, . . . ,m+n − k) (3.32). The figure represents the case m = 11, n = 5, ℓ = 4, k = 2 (x1, x2, x3, x4) =(2, 4, 5, 7), (y1, y2, y3, y4, y5, y6) = (2, 4, 7, 8, 9, 11).

The second type of mixed partition functions we use in this paper is (Figure 9)

ZDBm,n,k(uk+1, . . . , un|u1, . . . , uk|x1, . . . , xℓ|y1, . . . , yℓ+k)

=m〈y1 · · · yℓ+k|D(un) · · ·D(uk+1)B(uk) · · ·B(u1)|x1 · · · xℓ〉m, (3.31)

where the order of the layers of the B-operators and those of the D-operators is reversed fromthe first type of mixed partition functions (3.28). One can write the explicit form of (3.31)exactly as the skew Grothendieck polynomials. Consider the following partition functionsconstructed only from the B-operators (3.3) (Figure 10)

Zm+n−k,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+k,m+ 1, . . . ,m+ n− k)

=m〈y1 · · · yℓ+k,m+ 1 · · ·m+ n− k|B(m+n−k)(un) · · ·B(m+n−k)(u1)|x1 · · · xℓ〉m, (3.32)

where

B(m+n−k)(u) = a〈0|Ra,m+n−k(u) · · ·Ra1(u)|1〉a ∈ End(W1 ⊗ · · · ⊗Wm+n−k). (3.33)

We can also show that the mixed partition functions (3.31) are the same with the matrixelements (3.32).

Lemma 3.4. The following relation holds:


=ZDBm,n,k(uk+1, . . . , un|u1, . . . , uk|x1, . . . , xℓ|y1, . . . , yℓ+k). (3.34)

In Appendix B, we show a proof of (3.34) using equations only. A graphical derivation isgiven in Appendix C.

21

Applying the correspondence (3.21) to the partition functionsZm+n−k,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+k,m+1, . . . ,m+n−k) and combining with (3.34),we get


=G((m−ℓ−k)n−k ,λ)//µ(1− u−11 , . . . , 1− u−1

n ), (3.35)

where λj = yℓ−j+k+1−ℓ+j−k−1 (j = 1, . . . , ℓ+k) and µj = xℓ−j+1−ℓ+j−1 (j = 1, . . . , ℓ).

4 Grothendieck classes and K-theoretic pushforward

In this section, we use two types of partition functions of the integrable five-vertex modelintroduced in the last section and the Shigechi-Uchiyama commutation relation to show thattwo Grothendieck classes are directly connected via the K-theoretic pushforward. Let us firstrecall the algebraic geometry side [17, 18, 51, 52] again.

Let X be a smooth projective complex variety and K(X) the Grothendieck ring of isomor-phism classes of algebraic vector bundles over X. For a subvariety Y of X, the Grothendieckclass of the structure sheaf of Y is denoted [OY ], or [Y ] for short. Let A → X be a rank nvector bundle, and π : Grk(A) → X the Grassmann bundle of k-planes in A with universaltautological exact sequence 0 → S → π∗A → Q → 0 of vector bundles over Grk(A). Letσ = {σ1, . . . , σk} and ω = {ω1, . . . , ωn−k} be the set of Grothendieck roots for the subbundleS and the quotient bundle Q respectively, i.e., defined by the exterior powers of S and Q bythe relation [

∧i S] = ei(σ) and [∧j Q] = ej(ω), where ei(σ) and ej(ω) are the ith and jth

elementary symmetric functions with the set of variables σ and ω, respectively.Let f(σ;ω) be a Laurent polynomial in K(X)[σ±

i ;ω±j ] which are symmetric in σ variables

and also symmetric in ω variables, which makes it possible for the polynomial to be viewed as acertain Grothendieck class inK(Grk(A)). We consider the following K-theoretic pushforwardπ∗ : K(Grk(A)) → K(X) induced from π. By using quantum integrability, we show that thefollowing holds for the K-theoretic pushforward:

Theorem 4.1. Let A be a vector bundle on X of rank n, and π : Grk(A) → X the Grassmannbundle of k-planes in A with universal tautological exact sequence 0 → S → π∗A → Q → 0 ofvector bundles over Grk(A). Let α = {α1, . . . , αn}, σ = {σ1, . . . , σk} and ω = {ω1, . . . , ωn−k}be the sets of Grothendieck roots for bundles A, S and Q respectively. Then we have

π∗

( ∑

ν⊆(m−ℓ)ℓ

Gλ//ν(1− σ−11 , . . . , 1 − σ−1

k )G((m−ℓ)n−k ,ν)//µ(1− ω−11 , . . . , 1− ω−1

n−k)

)

=G((m−ℓ−k)n−k ,λ)//µ(1− α−11 , . . . , 1− α−1

n ), (4.1)

where ℓ is an integer satisfying 0 ≤ ℓ ≤ m − k and λ, µ and ν are partitions with λ ⊆(m− k − ℓ)k.

Proof. The following sum of products of the skew Grothendieck polynomials

g(σ;ω) =∑

ν⊆(m−ℓ)ℓ

Gλ//ν(1− σ−11 , . . . , 1− σ−1

k )G((m−ℓ)n−k ,ν)//µ(1− ω−11 , . . . , 1− ω−1

n−k),

(4.2)

22

is the right hand side of (3.29) under the following renaming of variables uj = σj , (j =1, . . . , k), uj = ωj−k, (j = k+1, . . . , n). By (3.29), we get the partition function representationof g(σ;ω)

g(σ;ω) =ZBDm,n,k(σ1, . . . , σk|ω1, . . . , ωn−k|x1, . . . , xℓ|y1, . . . , yℓ+k)

=m〈y1 · · · yℓ+k|B(σ1) · · ·B(σk)D(ω1) · · ·D(ωn−k)|x1 · · · xℓ〉m, (4.3)

where λj = yℓ−j+k+1 − ℓ + j − k − 1 (j = 1, . . . , ℓ + k) and µj = xℓ−j+1 − ℓ + j − 1 (j =1, . . . , ℓ). From the partition function representation (4.3), one can see for example thatfrom the commutativity relation (2.41) and (2.42), g(σ;ω) is symmetric in σ variables andalso in ω variables. We also note from the expressions of the matrix elements of the R-matrix for the integrable five-vertex model (2.5) that if one expands the partition functionrepresentation of g(σ;ω) in the σ and ω variables, the coefficient of each term is some integerwhich can be viewed as 1 = Gφ ∈ K(X) multiplied by the integer, hence g(σ;ω) is a Laurentpolynomial in K(X)[σ±

i ;ω±j ] which are symmetric in σ and in ω, and can be viewed as a

certain Grothendieck class in K(Grk(A)).To prove (4.1), one first applies the formula (1.1) for the K-theoretic pushforward to

g(σ;ω) and then use the partition function representation of g(σ;ω) (4.3) to get

π∗(g(σ;ω))

=∑

Snk∈([n]

k )

∏

i∈Snk,j∈Sn

k

1

1− αi/αjg(αSn

k;αSn

k)

=∑

Snk∈([n]

k )

∏

i∈Snk,j∈Sn

k

1

1− αi/αjm〈y1 · · · yℓ+k|

∏

i∈Snk

B(αi)∏

j∈Snk

D(αj)|x1 · · · xℓ〉m

= m〈y1 · · · yℓ+k|∑

Snk∈([n]

k )

∏

i∈Snk,j∈Sn

k

1

1− αi/αj

∏

i∈Snk

B(αi)∏

j∈Snk

D(αj)|x1 · · · xℓ〉m. (4.4)

Recall that Snk is a k-subset of [n] = {1, 2, . . . , n}, the set of k-subsets of n is denoted as

([n]k

),

and Snk = {1, 2, . . . , n}\Sn

k . Next, we use the commutation relation (2.43) to (4.4). Note that|x1 · · · xℓ〉m and m〈y1 · · · yℓ+k| which are vectors on the tensor product of (dual) quantumspaces W1 ⊗ · · · ⊗Wm (W ∗

1 ⊗ · · · ⊗W ∗m) do not disturb one to use the commutation relation.

Applying the commutation relation (2.43) to (4.4), we get

π∗(g(σ;ω)) = m〈y1 · · · yℓ+k|n∏

j=k+1

D(αj)k∏

j=1

B(αj)|x1 · · · xℓ〉m. (4.5)

The right hand side of (4.5) is nothing but the second type of the mixed partition functionsZDBm,n,k(αk+1, . . . , αn|α1, . . . , αk|x1, . . . , xℓ|y1, . . . , yℓ+k). Combining (4.5) with the corre-spondence formula (3.35), we find the pushforward of the Grothendieck class g(σ;ω) is givenby the skew Grothendieck polynomials

π∗(g(σ;ω)) = G((m−ℓ−k)n−k ,λ)//µ(1− α−11 , . . . , 1− α−1

n ). (4.6)

23

Note that in the proof above, what played the crucial role are matrix elements of thecommutation relation (1.2)


=∑

Snk∈([n]

k )

∏

i∈Snk,j∈Sn

k

1

1− ui/ujZBDm,n,k(uSn

k|uSn

k|x1, . . . , xℓ|y1, . . . , yℓ+k), (4.7)

where uSnk= {ui | i ∈ Sn

k } and uSnk= {uj | j ∈ Sn

k }. Using (3.29) and (3.35), this can be

written as the following identity for the skew Grothendieck polynomials.

Theorem 4.2. We have the following identity.

G((m−ℓ−k)n−k ,λ)//µ(1− u−11 , . . . , 1− u−1

n )

=∑

Snk∈([n]

k )

∏

i∈Snk,j∈Sn

k

1

1− ui/uj

∑

ν⊆(m−ℓ)ℓ

Gλ//ν(1− u−1Snk)G((m−ℓ)n−k ,ν)//µ(1− u−1

Snk

), (4.8)

where ℓ is an integer satisfying 0 ≤ ℓ ≤ m − k and λ, µ and ν are partitions with λ ⊆(m− k − ℓ)k, and 1− u−1

Snk= {1− u−1

i | i ∈ Snk } and 1− u−1

Snk

= {1− u−1j | j ∈ Sn

k }.

5 Special cases and application

Let us see in this section that certain special cases of Theorem 4.1 reproduce special cases ofthe formulas derived by Buch [17].

Let us consider ℓ = 0, µ = φ of (4.1) in Theorem 4.1. In this case, only the summandcorreseponding to ν = φ in g(σ;ω) (4.2) survives, and g(σ;ω) becomes the following productof two Grothendieck polynomials

g(σ;ω) = Gλ(1− σ−11 , . . . , 1− σ−1

k )G(m)n−k (1− ω−11 , . . . , 1− ω−1

n−k), (5.1)

where λ satisfies λ ⊆ (m − k)k. Specializing the pushforward formula (4.1) to this case, weget the following result:

π∗(Gλ(1− σ−11 , . . . , 1− σ−1

k )G(m)n−k (1− ω−11 , . . . , 1− ω−1

n−k))

= G((m−k)n−k ,λ)(1− α−11 , . . . , 1− α−1

n ). (5.2)

This is also special cases of the following K-theoretic pushforward formulas by Buch [17],Lemma 7.1 and Theorem 7.3 (see also [85, 86] for cohomological versions).

Lemma 5.1. (Buch [17], Lemma 7.1) Let A be a vector bundle of rank n over a variety X.Let π : P∗(A) → X be the dual projective bundle of A and Q be the tautological quotient ofπ∗A. Then we have

π∗(Gm(Q)) = Gm−n+1(A), (5.3)

for any integer m.

24

Theorem 5.2. (Buch [17], Theorem 7.3) Let A and B be vector bundles on X of rank n andℓ respectively. Let π : Grk(A) → X the Grassmann bundle of k-planes in A with universaltautological exact sequence 0 → S → π∗A → Q → 0 of vector bundles over Grk(A). Letα = {α1, . . . , αn}, β = {β1, . . . , βℓ}, σ = {σ1, . . . , σk} and ω = {ω1, . . . , ωn−k} be the setsof Grothendieck roots for bundles A, B, S and Q respectively. Let I = (I1, . . . , In−k) andJ = (J1, J2, . . . ) be sequences of integers satisfying Ij ≥ ℓ for all j. Then we have

π∗(GI(Q− B)GJ(S − B)) = GI−(k)n−k ,J(A−B). (5.4)

Here, GI(A−B) in (5.3), (5.4) are certain Grothendieck classes of K(X) extended to se-quences of integers [17] from the the case when I are partitions I = λ = (λ1, . . . , λi) satisfyingn ≥ i, in which case Gλ(A− B) are the Grassmannian double Grothendieck polynomials

Gλ(A− B) = Gλ(1− α−11 , . . . , 1− α−1

n |1− β1, . . . , 1 − βℓ). (5.5)

The Grassmannian double Grothendieck polynomials [15, 16, 17, 18, 19, 82] has the followingdeterminant form

Gλ(z1, . . . , zn|v1, . . . , vℓ) =det([zi|v]

λj+n−j(1− zi)j−1)1≤i,j≤n∏

1≤i<j≤n(zi − zj). (5.6)

where λ = (λ1, λ2, . . . , λn) is a partition, {z1, . . . , zn} and v = {v1, v2, . . . , vℓ} are sets ofvariables and

[zi|v]j = (zi ⊕ v1)(zi ⊕ v2) · · · (zi ⊕ vj), (5.7)

where z ⊕ v := z + v − zv.The case λ = φ, k = n− 1 of (5.2)

π∗(Gm(1− ω−11 )) = Gm−n+1(1− α−1

1 , . . . , 1− α−1n ), (5.8)

which corresponds to the case when the rank of the quotient bundle Q is 1, is nothing but(5.3) in Lemma 5.1 (Buch [17], Lemma 7.1) when m is positive.

We can also see that (5.2) is the special case ℓ = 0, I = (m)n−k, J = λ of (5.4) in Theorem5.2 (Buch [17], Theorem 7.3), which is explicitly

π∗(G(m)n−k (Q)Gλ(S)) = G(m)n−k−(k)n−k ,λ(A) = G((m−k)n−k ,λ)(A). (5.9)

Let us also see that Theorem 4.2 is a skew generalization of an identity for the Grothendieckpolynomials recently found by Guo and Sun [75], which is a generalization of the one forSchur polynomials by Feher, Nemethi and Rimanyi [76]. In fact, ℓ = 0, µ = φ of (4.8) is

G(mn−k ,λ)(1− u−11 , . . . , 1 − u−1

n )

=∑

Snk∈([n]

k )

∏

i∈Snk,j∈Sn

k

1

1− ui/ujGλ(1− u−1

Snk)Gmn−k (1− u−1

Snk

). (5.10)

It is easy to see from the determinant representation (3.5) that Gmn−k(1 − u−1Snk

) has the

factorized form

Gmn−k(1− u−1Snk

) =∏

j∈Snk

(1− u−1j )m. (5.11)

25

Inserting (5.11) into (5.10), we get

G(mn−k ,λ)(1− u−11 , . . . , 1− u−1

n ) =∑

Snk∈([n]

k )

Gλ(1− u−1Snk)

∏

j∈Snk

(1− u−1j )m

∏

i∈Snk,j∈Sn

k

(1− ui/uj). (5.12)

Using the z-variables zj = 1− u−1j , j = 1, . . . , n, (5.12) can be rewritten as

G(mn−k ,λ)(z1, . . . , zn) =∑

Snk∈([n]

k )

Gλ(zSnk)

∏

i∈Snk

(1− zi)n−k

∏

j∈Snk

zmj

∏

i∈Snk,j∈Sn

k

(zj − zi). (5.13)

This is the β = −1 and nonfactorial case of Theorem 3.1 in [75] by Guo and Sun.Finally, let us discuss an application of the pushforward formula. There are studies on

deriving algebraic formulas and identities by investigating the pushforward map in variousways and combining them. One example is the geometric derivation of the Jacobi-Trudi typeformulas [71, 74].

Let us derive an integration formula (residue formula) by combining the pushforwardformula which we derived with another expression which is obtained by applying the followingformula.

Proposition 5.3. (Allman [52], Prop 6.3) Let A be a vector bundle on X of rank n, andπ : Grk(A) → X the Grassmann bundle of k-planes in A with universal tautological exactsequence 0 → S → π∗A → Q → 0 of vector bundles over Grk(A). Let α = {α1, . . . , αn},σ = {σ1, . . . , σk} and ω = {ω1, . . . , ωn−k} be the sets of Grothendieck roots for bundles A, Sand Q respectively. Let f(σ;ω) be a Grothendieck class in K(Grk(A)), and z = {z1, . . . , zn}an alphabet of ordered, commuting variables. If f does not have poles in K(X) aside fromzero and the point at infinity, we have

π∗(f(σ;ω)) =1

(2πi)n

∮

C· · ·

∮

Cf(z)

∏

1≤i<j≤n

(1− zj/zi)

n∏

i,j=1

(zi/αj − 1)

n∏

i=1

dzizi

, (5.14)

where f(z) = f(z1, . . . , zk; zk+1, . . . , zn) and the integration contour C surrounds αj , j =1, . . . , n.

Note that (5.14) is equivalent to the expression in [52], which is written in the form oftaking residues at zero and the point at infinity. Applying the integration formula (5.14) to

26

g(σ;ω) (4.2), we get

π∗

( ∑

ν⊆(m−ℓ)ℓ

Gλ//ν(1− σ−11 , . . . , 1 − σ−1

k )G((m−ℓ)n−k ,ν)//µ(1− ω−11 , . . . , 1− ω−1

n−k)

)

=1

(2πi)n

∑

ν⊆(m−ℓ)ℓ

∮

C· · ·

∮

CGλ//ν(1− z−1

1 , . . . , 1− z−1k )

×G((m−ℓ)n−k ,ν)//µ(1− z−1k+1, . . . , 1− z−1

n )

∏

1≤i<j≤n

(1− zj/zi)

n∏

i,j=1

(zi/αj − 1)

n∏

i=1

dzizi

. (5.15)

Combining (4.1) and (5.15), we get the following.

Corollary 5.4. We have the following identity.

G((m−ℓ−k)n−k ,λ)//µ(1− α−11 , . . . , 1− α−1

n )

=1

(2πi)n

∑

ν⊆(m−ℓ)ℓ

∮

C· · ·

∮

CGλ//ν(1− z−1

1 , . . . , 1− z−1k )

×G((m−ℓ)n−k ,ν)//µ(1− z−1k+1, . . . , 1− z−1

n )

∏

1≤i<j≤n

(1− zj/zi)

n∏

i,j=1

(zi/αj − 1)

n∏

i=1

dzizi

, (5.16)

where ℓ is an integer satisfying 0 ≤ ℓ ≤ m − k and λ, µ and ν are partitions with λ ⊆(m− k − ℓ)k.

Taking ℓ = 0, µ = φ in (5.16) gives

G((m−k)n−k ,λ)(1− α−11 , . . . , 1− α−1

n )

=1

(2πi)n

∮

C· · ·

∮

CGλ(1− z−1

1 , . . . , 1 − z−1k )

n∏

j=k+1

(1− z−1j )m

∏

1≤i<j≤n

(1− zj/zi)

n∏

i,j=1

(zi/αj − 1)

n∏

i=1

dzizi

,

(5.17)

by using Gmn−k (1 − z−1k+1, . . . , 1 − z−1

n ) =

n∏

j=k+1

(1 − z−1j )m. Further, setting m = k, λ = φ

and using Gφ = 1, (5.17) becomes

1 =1

(2πi)n

∮

C· · ·

∮

C

n∏

j=k+1

(1− z−1j )k

∏

1≤i<j≤n

(1− zj/zi)

n∏

i,j=1

(zi/αj − 1)

n∏

i=1

dzizi

. (5.18)

27

Acknowledgments

We appreciate the referee for careful reading and numerous invaluable comments and sugges-tions to improve the paper. This work was partially supported by grant-in-Aid for ScientificResearch (C) No. 18K03205 and No. 16K05468.

Appendix A. On skew Grothendieck polynomials

Figure 11: The skew shape (5, 3, 3, 1)/(4, 3, 2) are boxes which are shaded in the left figure.Removing a box whose coordinate is (1, 4), (2, 3) or (3, 2) from the Young diagram (4, 3, 2) stillyields a Young diagram again. Adding these boxes to (5, 3, 3, 1)/(4, 3, 2) gives the extendedskew shape (5, 3, 3, 1)//(4, 3, 2), which are boxes shaded in the right figure.

In this appendix, we record a formulation of the skew Grothendieck polynomials Gλ//µ byYeliussizov [84], which was originally introduced by Buch [18]. Note that in this Appendix,we respect the notations in [84] as possible as we can, so the meaning of the A-operator, u-and x-variables are different from the ones used in the main part of this paper.

We first prepare notations. A partition is a nonincreasing sequence of nonnegative integersλ = (λ1, λ2, . . . , λℓ(λ), λℓ(λ)+1 = 0, . . . ), λ1 ≥ λ2 ≥ · · · ≥ λℓ(λ) > 0 with only finitely manynonzero terms. ℓ(λ) is called the length of the partition. We define |λ| for a partition λ as|λ| = λ1 + λ2 + · · ·+ λℓ(λ). We identify a partition λ with a Young diagram which is the set{(i, j) | 1 ≤ i ≤ ℓ(λ), 1 ≤ j ≤ λi}.

For a Young diagram µ, we say that a box in µ is removable if removing it yields a Youngdiagram of a partition. Let I(µ) be the set of removable boxes of µ. For two partitions λand µ satisfying µ ⊂ λ, we define λ//µ as λ//µ := λ/µ∪ I(µ). This is an extension of a skewshape λ/µ. We define a(λ//µ) as the number of columns of λ//µ that are not columns ofλ/µ. We say that a skew shape λ/µ is a horizontal strip if |λ/µ| is equal to the number ofcolumns of λ/µ.

For example, let us take λ and µ as λ = (5, 3, 3, 1), µ = (4, 3, 2) (Figure 11). I((4, 3, 2))consists of boxes whose coordinates are (1, 4), (2, 3), (3, 2). There are boxes which are inthe second and fourth column of (5, 3, 3, 1)//(4, 3, 2) but not in (5, 3, 3, 1)/(4, 3, 2), hencea((5, 3, 3, 1)//(4, 3, 2)) = 2.

28

Consider the free Z-module ZP = ⊕λ Z · λ with a basis of all partitions. Let u =(u1, u2, . . . ) and d = (d1, d2, . . . ) be sets of linear operators on ZP , which act on bases as

ui · λ =

{λ+� in column i, if possible,0, otherwise,

(A.1)

di · λ =

{λ−� in column i, if possible,0, otherwise.

(A.2)

Let β be a scalar parameter and we consider the free Z[β]-module Z[β]P = ⊕λZ[β] · λwith a basis of all partitions. We define u = (u1, u2, . . . ) as linear operators which act onZ[β]P as ui := ui − βuidi.

Let x be an indeterminate commuting with u and d, and define A(x) as

A(x) = · · · (1 + xu2)(1 + xu1). (A.3)

Let 〈·, ·〉 be a bilinear pairing on Z[β]P given by 〈λ, µ〉 = δλµ. For partitions λ and µ, the

skew Grothendieck polynomials Gβλ//µ(x1, . . . , xn) is defined in [84], Definition 4.1 as

Gβλ//µ(x1, . . . , xn) := 〈A(xn) · · ·A(x1) · µ, λ〉. (A.4)

The following is shown in [84].

Proposition A.1. (Yeliussizov [84], Proposition 4.3)For a single variable x, we have

Gβλ//µ(x) =

{(1− βx)a(λ//µ)x|λ/µ|, λ/µ is a horizontal strip,0, otherwise.

(A.5)

Noting that (1−u−1)∑ℓ+1

j=1 λj−∑ℓ

j=1 µj∏ℓ

j=1{u−1+ δλj+1,µj

(1−u−1)} in (3.8) can be rewrittenas

(1 − u−1)∑ℓ+1

j=1 λj−∑ℓ

j=1 µj

ℓ∏

j=1

{u−1 + δλj+1,µj(1− u−1)}

=(1 − u−1)|λ|−|µ|(1− (1− u−1))#{j|λj+1 6=µj}

=(1 − u−1)|λ/µ|(1− (1− u−1))a(λ//µ), (A.6)

and the interlacing relation λ ≻ µ is another way of saying that λ/µ is a horizontal strip,we find that the expression of the skew Grothendieck polynomials (3.8) which we defined asmatrix elements of the B-operator (3.7), is the same with the β = 1 case of (A.5 ) under thesubstitution x = 1− u−1.

(A.5 ) and the n = 1 case of (A.4 ) means

A(x) · µ =∑

λ≻µ

Gβλ//µ(x) · λ, (A.7)

29

and applying this relation repeatedly to the right hand side of (A.4 ) for generic n, we get

Gβλ//µ(x1, . . . , xn) =

∑

λ=λ(0)≻λ(1)≻···≻λ(n)=µ

n∏

j=1

Gβ

λ(j−1)//λ(j)(xj). (A.8)

Comparing with (3.18) in section 3, we note that the skew Grothendieck polynomialsGλ//µ(1−

u−11 , . . . , 1− u−1

n ) which we defined and has partition function expressions using B-operators(3.21) is the same with the β = 1 case of the one (A.4 ) which are introduced using Schur op-erators, where we substitute the u-variables in the main part of this paper into the x-variablesin this Appendix by xj = 1− u−1

j , j = 1, . . . , n.

Appendix B. Details of proofs of Lemma 3.3 and Lemma 3.4

In this appendix, we show Lemma 3.3 and Lemma 3.4 using equations only. We abbreviateR(u, 1) as R(u) in this Appendix.

Let us first show (3.26). For j = 1, . . . , n − 1, let ǫjm+n be a sequence of 0s and 1s

ǫjm+n = {ǫj1, ǫ

j2, . . . , ǫ

jm+n}, ǫ

j1, ǫ

j2, . . . , ǫ

jm+n = 0, 1, and define |ǫjm+n〉m+n and m+n〈ǫ

jm+n| as

|ǫjm+n〉m+n := |ǫj1〉1 ⊗ |ǫj2〉2 ⊗ · · · ⊗ |ǫjm+n〉m+n, (B.1)

m+n〈ǫjm+n| := 1〈ǫ

j1| ⊗ 2〈ǫ

j2| ⊗ · · · ⊗ m+n〈ǫ

jm+n|. (B.2)

We also define |ǫjm〉m and m〈ǫjm| similarly.We write the identity operator on W1 ⊗ · · · ⊗Wm+n as

Id =∑

ǫjm+n

|ǫjm+n〉m+n m+n〈ǫjm+n|, (B.3)

where∑

ǫjm+n

means∑

ǫj1,...,ǫjm+n=0,1

. We insert (B.3 ) between m+n〈y1 · · · yℓ,m + 1, . . . ,m +

n|, B(m+n)(uj), j = 1, . . . , n and |x1 · · · xℓ〉m+n, and decompose the partition functionsZm+n,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ,m+ 1, . . . ,m+ n) (3.24) as


=m+n〈y1 · · · yℓ,m+ 1, . . . ,m+ n|n∏

j=1

B(m+n)(uj)|x1 · · · xℓ〉m+n

=∑

ǫ1m+n,ǫ

2m+n,...,ǫ

n−1m+n

m+n〈y1 · · · yℓ,m+ 1, . . . ,m+ n|B(m+n)(u1)|ǫ1m+n〉m+n

×

n−1∏

j=2

m+n〈ǫj−1m+n|B

(m+n)(uj)|ǫjm+n〉m+n × m+n〈ǫ

n−1m+n|B

(m+n)(un)|x1 · · · xℓ〉m+n. (B.4)

First, we examine the factor m+n〈y1 · · · yℓ,m + 1, . . . ,m + n|B(m+n)(u1)|ǫ1m+n〉m+n in the

right hand side of (B.4 ). We use the explicit expression for the B-operator using R-matrices

30

B(m+n)(u1) = a〈0|Ra,m+n(u1) · · ·Ra1(u1)|1〉a and rewrite the factor as


=a〈0| ⊗ m〈y1 · · · yℓ| ⊗ m+1〈1| ⊗ · · · ⊗ m+n〈1|Ra,m+n(u1) · · ·Ra1(u1)

× |1〉a ⊗ |ǫ1m〉m ⊗ |ǫ1m+1〉m+1 ⊗ · · · ⊗ |ǫ1m+n〉m+n. (B.5)

We then write the identity operator on Wa ⊗W1 ⊗ · · · ⊗Wm+n as

Id =∑

γ1,ǫ1,...,ǫm+n=0,1

|γ1〉a ⊗ |ǫ1〉1 ⊗ · · · ⊗ |ǫm+n〉m+n a〈γ1| ⊗ 1〈ǫ1| ⊗ · · · ⊗ m+n〈ǫm+n|, (B.6)

and insert between Ra,m+1(u1) and Ram(u1) in (B.5 ) as


=∑

γ1,ǫ1,...,ǫm+n=0,1

a〈0| ⊗ m〈y1 · · · yℓ| ⊗ m+1〈1| ⊗ · · · ⊗ m+n〈1|Ra,m+n(u1) · · ·Ra,m+1(u1)

× |γ1〉a ⊗ |ǫ1〉1 ⊗ · · · ⊗ |ǫm+n〉m+n a〈γ1| ⊗ 1〈ǫ1| ⊗ · · · ⊗ m+n〈ǫm+n|Ram(u1) · · ·Ra1(u1)

× |1〉a ⊗ |ǫ1m〉m ⊗ |ǫ1m+1〉m+1 ⊗ · · · ⊗ |ǫ1m+n〉m+n. (B.7)

Since the product of R-matrices Ram(u1) · · ·Ra1(u1) act nontrivially on Wa, W1 . . . ,Wm andas identity on Wm+1 . . . ,Wm+n, we have

a〈γ1| ⊗ 1〈ǫ1| ⊗ · · · ⊗ m+n〈ǫm+n|Ra,m(u1) · · ·Ra1(u1)|1〉a ⊗ |ǫ1m〉m ⊗ |ǫ1m+1〉m+1 ⊗ · · · ⊗ |ǫ1m+n〉m+n

=m+n∏

j=m+1

δǫjǫ1j a〈γ1| ⊗ 1〈ǫ1| ⊗ · · · ⊗ m〈ǫm|Ram(u1) · · ·Ra1(u1)|1〉a ⊗ |ǫ1m〉m, (B.8)

whereRaj(u1), j = 1, . . . ,m on the right hand side are operators acting onWa⊗W1⊗· · ·⊗Wm.Inserting (B.8 ) into (B.7 ), one gets


=∑

γ1,ǫ1,...,ǫm=0,1


× |γ1〉a ⊗ |ǫ1〉1 ⊗ · · · ⊗ |ǫm〉m ⊗ |ǫ1m+1〉m+1 ⊗ · · · ⊗ |ǫ1m+n〉m+n

× a〈γ1| ⊗ 1〈ǫ1| ⊗ · · · ⊗ m〈ǫm|Ram(u1) · · ·Ra1(u1)|1〉a ⊗ |ǫ1m〉m. (B.9)

Since the product Ra,m+n(u1) · · ·Ra,m+1(u1) acts nontrivially on Wa, Wm+1, . . . ,Wm+n andas identity on W1, . . . ,Wm, we get


× |γ1〉a ⊗ |ǫ1〉1 ⊗ · · · ⊗ |ǫm〉m ⊗ |ǫ1m+1〉m+1 ⊗ · · · ⊗ |ǫ1m+n〉m+n

=a〈0| ⊗ m+1〈1| ⊗ · · · ⊗ m+n〈1|Ra,m+n(u1) · · ·Ra,m+1(u1)|γ1〉a ⊗ |ǫ1m+1〉m+1 ⊗ · · · ⊗ |ǫ1m+n〉m+n,(B.10)

if {ǫ1, . . . , ǫm} satisfies |ǫ1〉1 ⊗ · · · ⊗ |ǫm〉m = |y1 · · · yℓ〉m (1〈ǫ1| ⊗ · · · ⊗ m〈ǫm| = m〈y1 · · · yℓ|),and


× |γ1〉a ⊗ |ǫ1〉1 ⊗ · · · ⊗ |ǫm〉m ⊗ |ǫ1m+1〉m+1 ⊗ · · · ⊗ |ǫ1m+n〉m+n = 0, (B.11)

31

otherwise. Then we find (B.9 ) can be rewritten as


=∑

γ1=0,1

a〈0| ⊗ m+1〈1| ⊗ · · · ⊗ m+n〈1|Ra,m+n(u1) · · ·Ra,m+1(u1)

× |γ1〉a ⊗ |ǫ1m+1〉m+1 ⊗ · · · ⊗ |ǫ1m+n〉m+n

× a〈γ1| ⊗ m〈y1 · · · yℓ|Ram(u1) · · ·Ra1(u1)|1〉a ⊗ |ǫ1m〉m. (B.12)

The factor a〈0| ⊗ m+1〈1| ⊗ · · · ⊗ m+n〈1|Ra,m+n(u1) · · ·Ra,m+1(u1)|γ1〉a ⊗ |ǫ1m+1〉m+1 ⊗ · · · ⊗|ǫ1m+n〉m+n can be explicitly expressed using R-matrix elements as

a〈0| ⊗ m+1〈1| ⊗ · · · ⊗ m+n〈1|Ra,m+n(u1) · · ·Ra,m+1(u1)|γ1〉a ⊗ |ǫ1m+1〉m+1 ⊗ · · · ⊗ |ǫ1m+n〉m+n

=∑

γ2,...,γn=0,1

[R(u1)]01γnǫ1m+n

n−1∏

j=1

[R(u1)]γj+1,1

γj ,ǫ1m+j

. (B.13)

Inserting (B.13 ) into (B.12 ) gives


=∑

γ1,...,γn=0,1

[R(u1)]01γnǫ1m+n

n−1∏

j=1

[R(u1)]γj+1,1

γj ,ǫ1m+j

×a〈γ1| ⊗ m〈y1 · · · yℓ|Ram(u1) · · ·Ra1(u1)|1〉a ⊗ |ǫ1m〉m. (B.14)

Since [R(u1)]01γnǫ1m+n

= 1 if γn = 1, ǫ1m+n = 0 and [R(u1)]01γnǫ1m+n

= 0 otherwise, we can restrict

the sum over γn in the right hand side of (B.14 ) to γn = 1. Then we find the factor[R(u1)]

γn1γn−1ǫ1m+n−1

becomes [R(u1)]11γn−1ǫ1m+n−1

, which is equal to 1 if γn−1 = ǫ1m+n−1 = 1 and

vanishes otherwise. Continuing this argument, we find that we only need to consider the casewhich ǫ1m+1 = · · · = ǫ1m+n−1 = 1, ǫ1m+n = 0 is satisfied since otherwise m+n〈y1 · · · yℓ,m +

1, . . . ,m+ n|B(m+n)(u1)|ǫ1m+n〉m+n = 0, and we can restrict the sum in the right hand side

of (B.14 ) to the summand corresponding to γ1 = · · · = γn = 1. Then we find the right handside of (B.14 ) becomes

[R(u1)]0110([R(u1)]

1111)

n−1a〈1| ⊗ m〈y1 · · · yℓ|Ram(u1) · · ·Ra1(u1)|1〉a ⊗ |ǫ1m〉m

=m〈y1 · · · yℓ|D(m)(u1)|ǫ

1m〉m, (B.15)

hence we conclude

m+n〈y1 · · · yℓ,m+ 1, . . . ,m+ n|B(m+n)(u1)|ǫ1m+n〉m+n = m〈y1 · · · yℓ|D

(m)(u1)|ǫ1m〉m, (B.16)

if ǫ1m+n satisfies ǫ1m+1 = · · · = ǫ1m+n−1 = 1, ǫ1m+n = 0, and m+n〈y1 · · · yℓ,m + 1, . . . ,m +

n|B(m+n)(u1)|ǫ1m+n〉m+n = 0 otherwise.

Next, we examine m+n〈ǫ1m+n|B

(m+n)(u2)|ǫ2m+n〉m+n under ǫ1m+1 = · · · = ǫ1m+n−1 = 1,

ǫ1m+n = 0. We first write this explicitly as

m+n〈ǫ1m+n|B

(m+n)(u2)|ǫ2m+n〉m+n =

∑

γ1,...,γn=0,1

[R(u2)]00γnǫ2m+n

n−1∏

j=1

[R(u2)]γj+1,1

γj ,ǫ2m+j

×a〈γ1| ⊗ m〈ǫ1m|Ram(u2) · · ·Ra1(u2)|1〉a ⊗ |ǫ2m〉m, (B.17)

32

in the same way we did as above. Then we observe that the factor [R(u2)]00γnǫ2m+n

in the right

hand side of (B.17 ) is equal to 1 if γn = ǫ2m+n = 0, and vanishes otherwise. Then we note

the matrix element [R(u2)]γn1γn−1ǫ2m+n−1

becomes [R(u2)]01γn−1ǫ2m+n−1

and find that we only need

to consider γn−1 = 1, ǫ2m+n−1 = 0, since otherwise the matrix element becomes 0. Continuing

this argument, we find m+n〈ǫ1m+n|B

(m+n)(u2)|ǫ2m+n〉m+n = 0 unless ǫ2m+1 = · · · = ǫ2m+n−2 =

1, ǫ2m+n−1 = ǫ2m+n = 0, and if ǫ2m+n satisfies ǫ2m+1 = · · · = ǫ2m+n−2 = 1, ǫ2m+n−1 = ǫ2m+n = 0,we can restrict the sum in the right hand side of (B.17 ) to the summand corresponding toγ1 = · · · = γn−1 = 1, γn = 0 and we have

m+n〈ǫ1m+n|B

(m+n)(u2)|ǫ2m+n〉m+n =[R(u2)]

0000[R(u2)]

0110([R(u2)]

1111)

n−2

×a〈1| ⊗ m〈ǫ1m|Ram(u2) · · ·Ra1(u2)|1〉a ⊗ |ǫ2m〉m

=m〈ǫ1m|D(m)(u2)|ǫ2m〉m. (B.18)

We continue this argument and show the following for j = 2, . . . , n − 1: for ǫj−1m+n such

that ǫj−1m+1 = · · · = ǫj−1

m+n−j+1 = 1, ǫj−1m+n−j+2 = · · · = ǫj−1

m+n = 0, the following relation holds

m+n〈ǫj−1m+n|B

(m+n)(uj)|ǫjm+n〉m+n = m〈ǫj−1

m |D(m)(uj)|ǫjm〉m, (B.19)

if ǫjm+n satisfies ǫjm+1 = · · · = ǫjm+n−j = 1, ǫjm+n−j+1 = · · · = ǫjm+n = 0. Otherwise,

m+n〈ǫj−1m+n|B

(m+n)(uj)|ǫjm+n〉m+n = 0.

One can also show in a similar way that for ǫn−1m+n such that ǫn−1

m+1 = 1, ǫn−1m+2 = · · · =

ǫn−1m+n = 0, the following relation holds

m+n〈ǫn−1m+n|B

(m+n)(un)|x1 · · · xℓ〉m+n = m〈ǫn−1m |D(m)(un)|x1 · · · xℓ〉m. (B.20)

From the arguments above and using (B.16 ), (B.19 ) and (B.20 ), the right hand side of(B.4 ) can be rewritten in the following way:

∑

ǫ1m+n,ǫ

2m+n,...,ǫ

n−1m+n


×

n−1∏

j=2

m+n〈ǫj−1m+n|B

(m+n)(uj)|ǫjm+n〉m+n × m+n〈ǫ

n−1m+n|B

(m+n)(un)|x1 · · · xℓ〉m+n

=∑

ǫ1m,ǫ2m,...,ǫn−1

m

m〈y1 · · · yℓ|D(m)(u1)|ǫ

1m〉m

×

n−1∏

j=2

m〈ǫj−1m |D(m)(uj)|ǫ

jm〉m × m〈ǫn−1

m |D(m)(un)|x1 · · · xℓ〉m

=m〈y1 · · · yℓ|n∏

j=1

D(m)(uj)|x1 · · · xℓ〉m

=Um,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ), (B.21)

hence we conclude (3.26) holds. Note that we used the expression for the identity operatoracting on W1 ⊗ · · · ⊗Wm

Id =∑

ǫjm

|ǫjm〉m m〈ǫjm|, (B.22)

33

in the second equality in (B.21 ). We also remark that from the argument above, ǫjk,

j = 1, . . . , n − 1, k = m + 1, . . . ,m + n are fixed to ǫjm+1 = · · · = ǫjm+n−j = 1, ǫjm+n−j+1 =

· · · = ǫjm+n = 0 and the sum∑

ǫ1m+n,ǫ

2m+n,...,ǫ

n−1m+n

is reduced to∑

ǫ1m,ǫ2m,...,ǫn−1

m

which is used in the

first equality in (B.21 ).

We can show (3.34) in the same way. For j = 1, . . . , n − 1, we define |ǫjm+n−k〉m+n−k :=

|ǫj1〉1⊗· · ·⊗|ǫjm+n−k〉m+n−k and m+n−k〈ǫjm+n−k| := 1〈ǫ

j1|⊗· · ·⊗m+n−k〈ǫ

jm+n−k| for a sequence

of 0s and 1s ǫjm+n−k = {ǫj1, ǫ

j2, . . . , ǫ

jm+n−k} (ǫj1, ǫ

j2, . . . , ǫ

jm+n−k = 0, 1), write the identity

operator on W1 ⊗ · · · ⊗Wm+n−k as

Id =∑

ǫjm+n−k

|ǫjm+n−k〉m+n−k m+n−k〈ǫjm+n−k|, (B.23)

and insert into Zm+n−k,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+k,m+ 1, . . . ,m+ n− k) (3.32) anddecompose as


=m〈y1 · · · yℓ+k,m+ 1 · · ·m+ n− k|B(m+n−k)(un) · · ·B(m+n−k)(u1)|x1 · · · xℓ〉m

=∑

ǫ1m+n−k

,...,ǫn−1m+n−k

m+n−k〈y1 · · · yℓ+k,m+ 1, . . . ,m+ n− k|B(m+n−k)(un)|ǫ1m+n−k〉m+n−k

×

n−1∏

j=2

m+n−k〈ǫj−1m+n−k|B

(m+n−k)(un−j+1)|ǫjm+n−k〉m+n−k

× m+n−k〈ǫn−1m+n−k|B

(m+n−k)(u1)|x1 · · · xℓ〉m+n−k. (B.24)

We analyze the factors in (B.24 ) in the following order.One first shows that if ǫ1m+n−k satisfies ǫ1m+1 = · · · = ǫ1m+n−k−1 = 1, ǫ1m+n−k = 0, the

following relation holds


=m〈y1 · · · yℓ+k|D(m)(un)|ǫ

1m〉m, (B.25)

and m+n−k〈y1 · · · yℓ+k,m+ 1, . . . ,m+ n− k|B(m+n−k)(un)|ǫ1m+n−k〉m+n−k = 0 otherwise.

Next, we show the following for j = 2, . . . , n − k: for ǫj−1m+n−k such that ǫj−1

m+1 = · · · =

ǫj−1m+n−k−j+1 = 1, ǫj−1

m+n−k−j+2 = · · · = ǫj−1m+n−k = 0, the following relation holds


(m+n−k)(un−j+1)|ǫjm+n−k〉m+n−k = m〈ǫj−1

m |D(m)(un−j+1)|ǫjm〉m, (B.26)

if ǫjm+n−k satisfies ǫjm+1 = · · · = ǫjm+n−k−j = 1, ǫjm+n−k−j+1 = · · · = ǫjm+n−k = 0. Otherwise,


(m+n−k)(un−j+1)|ǫjm+n−k〉m+n−k = 0.

The third step is to show the following for j = n− k+1, . . . , n− 1: for ǫj−1m+n−k such that

ǫj−1m+1 = · · · = ǫj−1

m+n−k = 0, the following relation holds


(m+n−k)(un−j+1)|ǫjm+n−k〉m+n−k = m〈ǫj−1

m |B(m)(un−j+1)|ǫjm〉m, (B.27)

34

if ǫjm+n−k satisfies ǫjm+1 = · · · = ǫjm+n−k = 0. Otherwise, we have


(m+n−k)(un−j+1)|ǫjm+n−k〉m+n−k = 0.

Finally, we show that for ǫn−1m+n−k such that ǫn−1

m+1 = · · · = ǫn−1m+n−k = 0, the following

relation holds

m+n−k〈ǫn−1m+n−k|B

(m+n−k)(u1)|x1 · · · xℓ〉m+n−k = m〈ǫn−1m |B(m)(u1)|x1 · · · xℓ〉m. (B.28)

From the arguments above and inserting (B.25 ), (B.26 ), (B.27 ) and (B.28 ) into the righthand side of (B.24 ) and using (B.22 ), we can show the right hand side of (B.24 ) can berewritten in the following way:

∑

ǫ1m+n−k

,...,ǫn−1m+n−k


×

n−1∏

j=2


(m+n−k)(un−j+1)|ǫjm+n−k〉m+n−k

× m+n−k〈ǫn−1m+n−k|B

(m+n−k)(u1)|x1 · · · xℓ〉m+n−k

=∑

ǫ1m,...,ǫn−1

m

m+n−k〈y1 · · · yℓ+k|D(m)(un)|ǫ

1m〉m

n−k∏

j=2

m〈ǫj−1m |D(m)(un−j+1)|ǫ

jm〉m

×n−1∏

j=n−k+1

m〈ǫj−1m |B(m)(un−j+1)|ǫ

jm〉m × m〈ǫn−1

m |B(m)(u1)|x1 · · · xℓ〉m

=〈y1 · · · yℓ+k|D(m)(un) · · ·D

(m)(uk+1)B(m)(uk) · · ·B

(m)(u1)|x1 · · · xℓ〉m

=ZDBm,n,k(uk+1, . . . , un|u1, . . . , uk|x1, . . . , xℓ|y1, . . . , yℓ+k), (B.29)

hence we conclude (3.34) holds. Note that ǫjk, j = 1, . . . , n−1, k = m+1, . . . ,m+n are fixed

to ǫjm+1 = · · · = ǫjm+n−k−j = 1, ǫjm+n−k−j+1 = · · · = ǫjm+n−k = 0 for j = 1, . . . , n − k − 1,

and ǫjm+1 = · · · = ǫjm+n−k = 0 for j = n− k, . . . , n− 1 by the argument above, and the sum∑

ǫ1m+n−k

,...,ǫn−1m+n−k

is reduced to∑

ǫ1m,...,ǫn−1

m

which is used in the first equality in (B.29 ).

Appendix C. Graphical derivations of Lemma 3.3 and Lemma

3.4

In this appendix, we give graphical derivations of Lemma 3.3 and Lemma 3.4.First, let us show (3.26). Figure 7 in section 3 represents the partition function

Zm+n,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ,m + 1, . . . ,m + n) (3.24), and by graphical observa-tion, one can see that only one configuration is allowed in the rightmost n columns, whichis graphically Figure 12. Let us explain this observation in more detail. First, we look theR-matrix at the upper right corner in Figure 7 and note that the output of the auxiliaryspace and the quantum space Wm+n is 0 and 1, respectively. From this, we note that wecan restrict the input of the auxiliary space and the quantum space Wm+n to 1 and 0 re-spectively, since [Ra,m+n(u1, 1)]

01γnǫ1m+n

6≡ 0 if and only if γn = 1 and ǫ1m+n = 0 (see Figure

35

Figure 12: A graphical description which shows Zm+n,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ,m+1, . . . ,m+ n) = Um,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ) (3.26).

1). [Ra,m+n(u1, 1)]01γnǫ1m+n

corresponds to the factor [R(u1)]01γnǫ1m+n

in (B.14 ) in the previ-

ous appendix. Next, we move to the second R-matrix on the top row, counted from right.The outputs of the auxiliary space and the quantum space Wm+n−1 are both 1, hence wefind the inputs of both spaces can be restricted to 1, since [Ra,m+n−1(u1, 1)]

11γn−1ǫ1m+n−1

is

not identically zero if and only if γn−1 = ǫ1m+n−1 = 1 (see Figure 1). Continuing this ob-servation, one can see that only one configuration is allowed in the rightmost n columns,and the part constructed from the remaining m columns is nothing but the partition func-tions Um,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ) (3.23) (Figure 6), which is the graphical meaningof Figure 12. The product of the R-matrix elements on the rightmost n columns is 1 since allR-matrix elements which appear in that part are either [R(uj , 1)]

0000 = 1, [R(uj , 1)]

1111 = 1 or

[R(uj , 1)]0110 = 1. Hence we conclude Zm+n,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ,m+1, . . . ,m+n)

is equal to Um,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ) multiplied by 1, and we get (3.26).

(3.34) can be graphically derived in a similar way. Figure 10 in section 3 represents theparition function Zm+n−k,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+k,m + 1, . . . ,m + n − k) (3.32),and by graphical observation, we find that only one configuration is allowed in the right-most n − k columns ( Figure 13). In more detail, we first perform graphical observationsstarting from the R-matrix at the upper right corner, and note that we can restrict theconfiguration of the rightmost n − k columns of the top n − k rows in the same way aswe have seen in the observation when going from Figure 7 to Figure 12. Next, one looksthe (n − k + 1)-th row counted from top. Since the outputs of the auxiliary space andthe quantum spaces Wm+1, . . . ,Wm+n−k are all 0, we can conclude that we can restrict allthe inputs of those spaces to 0, since [R(uk, 1)]

00j1j2

6≡ 0 if and only if j1 = j2 = 0, and

[R(uk, 1)]00j1j2

= 0 otherwise (see Figure 1). This observation applies to the remaining bottomk − 1 rows as well, and one can see that only one configuration is allowed in the right-most n − k columns, and the part constructed from the remaining m columns is nothing

36

Figure 13: A graphical description which showsZm+n−k,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+k,m + 1, . . . ,m + n − k) =ZDBm,n,k(uk+1, . . . , un|u1, . . . , uk|x1, . . . , xℓ|y1, . . . , yℓ+k) (3.34).

but the mixed partition functions ZDBm,n,k(uk+1, . . . , un|u1, . . . , uk|x1, . . . , xℓ|y1, . . . , yℓ+k)(3.31) (Figure 9), which is the graphical meaning of Figure 13. Since the R-matrix elementswhich appear in the rightmost n − k columns are either [R(uj , 1)]

0000 = 1, [R(uj, 1)]

1111 = 1

or [R(uj , 1)]0110 = 1, the product of all R-matrix elements in the rightmost n − k columns

is 1. Multiplying this 1 by ZDBm,n,k(uk+1, . . . , un|u1, . . . , uk|x1, . . . , xℓ|y1, . . . , yℓ+k) givesZm+n−k,n(u1, . . . , un|x1, . . . , xℓ|y1, . . . , yℓ+k,m+1, . . . ,m+ n− k), hence we conclude (3.34).

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