grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/slides/bandlow.pdf · gof...
TRANSCRIPT
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Grothendieck expansions of symmetricpolynomials
Jason Bandlow (joint work with Jennifer Morse)
University of Pennsylvania
August 3rd, 2010 – FPSACSan Francisco State University
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Outline
Symmetric functions
Tableaux–Schur expansions
Grothendieck functions and their dual basis
Main theorem
Examples
Sketch of proof
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The monomial basis
The monomial symmetric functions are indexed by partitions
λ = (λ1, λ2, . . . , λk) λi ≥ λi+1
mλ =∑α
xα
α a rearrangement of the parts of λ and infinitely many 0’s
Example
m2,1 =(x21x2 + x1x2
2 ) + (x1x23 + x2
1x3) + . . .
+ (x22x3 + x2x2
3 ) + . . .
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The complete homogeneous basis
The (complete) homogeneous symmetric functions are defined by
hi =∑λ`i
mλ
hλ = hλ1hλ2 . . . hλk
Example
h3 = m3 + m2,1 + m1,1,1
h4,2,1 = h4h2h1
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The Hall inner product
Defined by
〈hλ,mµ〉 =
{1 if λ = µ
0 otherwise
Proposition
If {fλ}, {f ∗λ } and {gλ}, {g∗λ} are two pairs of dual bases with
fλ =∑µ
Mλ,µgµ
then
g∗µ =∑λ
Mλ,µf ∗λ
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The Hall inner product
Defined by
〈hλ,mµ〉 =
{1 if λ = µ
0 otherwise
Proposition
If {fλ}, {f ∗λ } and {gλ}, {g∗λ} are two pairs of dual bases with
fλ =∑µ
Mλ,µgµ
then
g∗µ =∑λ
Mλ,µf ∗λ
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Semistandard Young tableaux
A left-and-bottom justified, partition-shaped array of numbers,weakly increasing across rows and strictly increasing up columns.
Example
7 75 6 6 83 3 4 71 1 2 2 2
Shape: (5, 4, 4, 2)Evaluation: (2, 3, 2, 1, 1, 2, 3, 1)Reading word: 775668334711222
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Semistandard Young tableaux
A left-and-bottom justified, partition-shaped array of numbers,weakly increasing across rows and strictly increasing up columns.
Example
7 75 6 6 83 3 4 71 1 2 2 2
Shape: (5, 4, 4, 2)Evaluation: (2, 3, 2, 1, 1, 2, 3, 1)Reading word: 775668334711222
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Knuth equivalence
An equivalence relation on words generated by
yxz ≡ yzx if x < y ≤ z
xzy ≡ zxy if x ≤ y < z
Key Fact
Every word is Knuth equivalent to the reading word of exactly onetableau.
Example
rw
(3 41 2 3
)= 34123 ≡ 31423 ≡ 31243 ≡ 13243 ≡ 13423
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The Schur basis
DefinitionThe Schur functions are given by
sλ =∑
T∈SSYT (λ)
xev(T )
Example
s2,1 = x21x2+ x1x2
2+ 2x1x2x3+ · · ·21 1
21 2
31 2
21 3 · · ·
Fact: Schur functions are a self-dual basis of the symmetricfunctions.
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The Schur basis
DefinitionThe Schur functions are given by
sλ =∑
T∈SSYT (λ)
xev(T )
Example
s2,1 = x21x2+ x1x2
2+ 2x1x2x3+ · · ·21 1
21 2
31 2
21 3 · · ·
Fact: Schur functions are a self-dual basis of the symmetricfunctions.
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The Schur basis
DefinitionThe Schur functions are given by
sλ =∑
T∈SSYT (λ)
xev(T )
Example
s2,1 = x21x2+ x1x2
2+ 2x1x2x3+ · · ·21 1
21 2
31 2
21 3 · · ·
Fact: Schur functions are a self-dual basis of the symmetricfunctions.
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The Schur basis
Using the fact that Schur functions are symmetric, we can rewritethe definition as
sλ =∑µ
Kλ,µmµ
where Kλ,µ is the number of semistandard tableaux of shape λ andevaluation µ.
Using the proposition about dual bases, we get
hµ =∑λ
Kλ,µsλ
which can be rewritten as
hµ =∑T∈Tµ
ssh(T )
where Tµ is the set of all semistandard tableaux of evaluation µ.
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The Schur basis
Using the fact that Schur functions are symmetric, we can rewritethe definition as
sλ =∑µ
Kλ,µmµ
where Kλ,µ is the number of semistandard tableaux of shape λ andevaluation µ.Using the proposition about dual bases, we get
hµ =∑λ
Kλ,µsλ
which can be rewritten as
hµ =∑T∈Tµ
ssh(T )
where Tµ is the set of all semistandard tableaux of evaluation µ.
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The Schur basis
Using the fact that Schur functions are symmetric, we can rewritethe definition as
sλ =∑µ
Kλ,µmµ
where Kλ,µ is the number of semistandard tableaux of shape λ andevaluation µ.Using the proposition about dual bases, we get
hµ =∑λ
Kλ,µsλ
which can be rewritten as
hµ =∑T∈Tµ
ssh(T )
where Tµ is the set of all semistandard tableaux of evaluation µ.
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Tableaux–Schur expansions
Elements of a family {fα} of symmetric functions havetableaux–Schur expansions if there exist sets Tα of semistandardtableaux and weight functions wtα such that
fα =∑T∈Tα
wtα(T )ssh(T )
Goal: find appropriate sets and modifications of wt so that
fα =∑S∈Sα
wtα(S)Gsh(S)
fα =∑R∈Rα
wtα(R)gsh(R)
where G and g are, respectively, the Grothendieck anddual-Grothendieck functions.
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Tableaux–Schur expansions
Elements of a family {fα} of symmetric functions havetableaux–Schur expansions if there exist sets Tα of semistandardtableaux and weight functions wtα such that
fα =∑T∈Tα
wtα(T )ssh(T )
Goal: find appropriate sets and modifications of wt so that
fα =∑S∈Sα
wtα(S)Gsh(S)
fα =∑R∈Rα
wtα(R)gsh(R)
where G and g are, respectively, the Grothendieck anddual-Grothendieck functions.
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Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
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Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
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Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
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Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
![Page 22: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/22.jpg)
Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
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Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
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Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
![Page 25: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/25.jpg)
Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
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Grothendieck polynomials
I Introduced by Lascoux-Schutzenberger (1982)
I Represent K -theory classes of structure sheaves of Schubertvarieties in GLn
I Given by elements of Z[[x1, . . . , xn]]
I Analogous to Schubert polynomials
I Sign-alternating by degree, equal to Schubert polynomial inbottom degree
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Stable Grassmannian Grothendieck functions
Stable:
I Due to Fomin-Kirillov
I Limit as number of variables →∞
Grassmannian:
I Indexed by partitions
I Power series of symmetric functions
I Sign-alternating by degree in the Schur basis
I Analog of Schur functions
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Stable Grassmannian Grothendieck functions
Stable:
I Due to Fomin-Kirillov
I Limit as number of variables →∞Grassmannian:
I Indexed by partitions
I Power series of symmetric functions
I Sign-alternating by degree in the Schur basis
I Analog of Schur functions
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Grassmannian Grothendieck functions
Theorem (Buch)
Gλ =∑
S∈SVT (λ)
ε(S)xev(S)
where SVT (λ) is the set of set-valued tableaux of shape λ.
Example
G1 = s1 −s1,1 +s1,1,1 . . .
1 12 123 · · ·
23 234 · · ·...
...
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Grassmannian Grothendieck functions
Theorem (Buch)
Gλ =∑
S∈SVT (λ)
ε(S)xev(S)
where SVT (λ) is the set of set-valued tableaux of shape λ.
Example
G1 = s1 −s1,1 +s1,1,1 . . .
1 12 123 · · ·
23 234 · · ·...
...
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Set-valued tableaux
Example
234 45
1 12 3
shape(S) = (3, 2) evaluation(S) = (2, 2, 2, 2, 1)
ε(S) = (−1)|ev(S)|+|sh(S)| = 1
From
Gλ =∑
S∈SVT (λ)
ε(S)xev(S)
we seeGλ = sλ ± higher degree terms
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Set-valued tableaux
Example
234 45
1 12 3
shape(S) = (3, 2) evaluation(S) = (2, 2, 2, 2, 1)
ε(S) = (−1)|ev(S)|+|sh(S)| = 1
From
Gλ =∑
S∈SVT (λ)
ε(S)xev(S)
we seeGλ = sλ ± higher degree terms
![Page 33: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/33.jpg)
Set-valued tableaux
Example
234 45
1 12 3
shape(S) = (3, 2) evaluation(S) = (2, 2, 2, 2, 1)
ε(S) = (−1)|ev(S)|+|sh(S)| = 1
From
Gλ =∑
S∈SVT (λ)
ε(S)xev(S)
we seeGλ = sλ ± higher degree terms
![Page 34: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/34.jpg)
Set-valued tableaux
We introduce the reading word of a set-valued tableau, defined by:
I Proceed row by row, from top to bottom
I In each row, first ignore the smallest element of each cell.Then read the remaining elements from right to left, and fromlargest to smallest within each cell.
I Read the smallest elements of each cell from left to right.
Example
234 45
1 12 3
has reading word: 543242113.
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Dual Grothendieck functionsWe denote the dual basis to the Grothendieck by g
Theorem (Lam, Pylyavskyy)
gλ =∑
R∈RPP(λ)
xev(R)
where RPP(λ) is the set of reverse plane partitions of shape λ.
Example
g2,1 = s2,1 +s2
21 1
11 1
21 3
11 2
......
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Dual Grothendieck functionsWe denote the dual basis to the Grothendieck by g
Theorem (Lam, Pylyavskyy)
gλ =∑
R∈RPP(λ)
xev(R)
where RPP(λ) is the set of reverse plane partitions of shape λ.
Example
g2,1 = s2,1 +s2
21 1
11 1
21 3
11 2
......
![Page 37: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/37.jpg)
Reverse plane partitions
Example
2 32 2 41 2 21 1 1 1
shape(R) = (4, 3, 3, 2) evaluation(R) = (4, 3, 1, 1)
From
gλ =∑
R∈RPP(λ)
xev(R)
we seegλ = sλ ± lower degree terms
![Page 38: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/38.jpg)
Reverse plane partitions
Example
2 32 2 41 2 21 1 1 1
shape(R) = (4, 3, 3, 2) evaluation(R) = (4, 3, 1, 1)
From
gλ =∑
R∈RPP(λ)
xev(R)
we seegλ = sλ ± lower degree terms
![Page 39: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/39.jpg)
Dual Grothendieck functions
We define the reading word of a reverse plane partition to be asubsequence of the usual reading word, where we only take thebottommost occurence of every letter in each column.
Example
2 32 2 41 2 21 1 1 1
has reading word 324221111.
![Page 40: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/40.jpg)
Generalizing tableaux–Schur expansions
Given any set Tα of semistandard tableaux, we definecorresponding sets Sα (respectively Rα) of set-valued tableaux(reverse plane partitions) defined by
S ∈ Sα ⇐⇒ rw(S) ≡ rw(T ) for some T ∈ Tα
R ∈ Rα ⇐⇒ rw(R) ≡ rw(T ) for some T ∈ Tα
Given a statistic wtα on Tα, we can extend it to reading words bywtα(w) = wtα(T ) if w ≡ rw(T ).
![Page 41: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/41.jpg)
Generalizing tableaux–Schur expansions
Given any set Tα of semistandard tableaux, we definecorresponding sets Sα (respectively Rα) of set-valued tableaux(reverse plane partitions) defined by
S ∈ Sα ⇐⇒ rw(S) ≡ rw(T ) for some T ∈ Tα
R ∈ Rα ⇐⇒ rw(R) ≡ rw(T ) for some T ∈ Tα
Given a statistic wtα on Tα, we can extend it to reading words bywtα(w) = wtα(T ) if w ≡ rw(T ).
![Page 42: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/42.jpg)
Examples
If T =
43 41 2 ∈ Tα, then
43 41 2
431 24
34 41 2
4 41 23
4
1 234
43412 43412 43412 44312 44312
are in Sα
and
43 41 2
4 41 31 2
4 43 41 2
44 41 31 2 · · ·
43412 44312 43412 44312
are in Rα. All of these will have the same weight.
![Page 43: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/43.jpg)
Examples
If T =
43 41 2 ∈ Tα, then
43 41 2
431 24
34 41 2
4 41 23
4
1 234
43412 43412 43412 44312 44312
are in Sα and
43 41 2
4 41 31 2
4 43 41 2
44 41 31 2 · · ·
43412 44312 43412 44312
are in Rα. All of these will have the same weight.
![Page 44: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/44.jpg)
Statement of main theorem
Theorem (B-Morse)
Let fα be a family of symmetric functions with
fα =∑T∈Tα
wtα(T )ssh(T ).
Then
fα =∑S∈Sα
ε(S)wtα(S)gsh(S)
and
fα =∑R∈Rα
wtα(R)Gsh(R).
![Page 45: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/45.jpg)
Complete homogeneous functions
We have a tableaux-Schur expansion of the homogeneous functionsby
hµ =∑T∈Tµ
ssh(T )
where Tµ is the set of all tableaux of evaluation µ. Thus we have
hµ =∑S∈Sµ
ε(S)gsh(S) =∑R∈Rµ
Gsh(R)
where Sµ is the set of all set-valued tableaux of evaluation µ andRµ is the set of all reverse plane partitions of evaluation µ.
![Page 46: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/46.jpg)
Example
h3,2 = s3,2 +s4,1 +s5
2 21 1 1
21 1 1 2 1 1 1 2 2
= g3,2 +g4,1 +g5
−g3,1 −g4
21 1 12 1 1 12 2
= G3,2 +G4,1 +G5
+2G3,2,1 +2G4,1,1 + · · ·22 21 1 1
21 21 1 1
221 1 1 2
211 1 1 2
![Page 47: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/47.jpg)
Example
h3,2 = s3,2 +s4,1 +s5
2 21 1 1
21 1 1 2 1 1 1 2 2
= g3,2 +g4,1 +g5
−g3,1 −g4
21 1 12 1 1 12 2
= G3,2 +G4,1 +G5
+2G3,2,1 +2G4,1,1 + · · ·22 21 1 1
21 21 1 1
221 1 1 2
211 1 1 2
![Page 48: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/48.jpg)
Example
h3,2 = s3,2 +s4,1 +s5
2 21 1 1
21 1 1 2 1 1 1 2 2
= g3,2 +g4,1 +g5
−g3,1 −g4
21 1 12 1 1 12 2
= G3,2 +G4,1 +G5
+2G3,2,1 +2G4,1,1 + · · ·22 21 1 1
21 21 1 1
221 1 1 2
211 1 1 2
![Page 49: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/49.jpg)
Hall-Littlewood functions
The Hall-Littlewood functions form a basis for the symmetricfunctions over the field Q(t). Interpretations as
I Deformation of Weyl character formula
I Graded Sn-character of certain cohomology rings
I Representation theory of groups of matrices over finite fields
among others.
We will use the version typically denoted by Hµ[X ; t] or Q ′µ(x ; t) inthe literature.
Example
H1,1,1[X ; t] = s1,1,1 + (t2 + t)s2,1 + t3s3
![Page 50: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/50.jpg)
Hall-Littlewood functions
The Hall-Littlewood functions form a basis for the symmetricfunctions over the field Q(t). Interpretations as
I Deformation of Weyl character formula
I Graded Sn-character of certain cohomology rings
I Representation theory of groups of matrices over finite fields
among others.
We will use the version typically denoted by Hµ[X ; t] or Q ′µ(x ; t) inthe literature.
Example
H1,1,1[X ; t] = s1,1,1 + (t2 + t)s2,1 + t3s3
![Page 51: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/51.jpg)
Hall-Littlewood functions
Theorem (Lascoux-Schutzenberger)
Hµ[X ; t] =∑T∈Tµ
tch(T )ssh(T )
where charge is a non-negative integer statistic defined on wordsand constant on Knuth equivalence classes.
![Page 52: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/52.jpg)
The charge statistic
Example
3 42 2 31 1 1 2
3 4 2 2 3 1 1 1 2
![Page 53: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/53.jpg)
The charge statistic
Example
3 42 2 31 1 1 2
0
3 4 2 2 3 1 1 1 2
![Page 54: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/54.jpg)
The charge statistic
Example
3 42 2 31 1 1 2
0 0
3 4 2 2 3 1 1 1 2
![Page 55: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/55.jpg)
The charge statistic
Example
3 42 2 31 1 1 2
0 0 0
3 4 2 2 3 1 1 1 2
![Page 56: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/56.jpg)
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0
3 4 2 2 3 1 1 1 2
![Page 57: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/57.jpg)
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0 0
3 4 2 2 3 1 1 1 2
![Page 58: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/58.jpg)
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0 0 0
3 4 2 2 3 1 1 1 2
![Page 59: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/59.jpg)
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0 1 0 0
3 4 2 2 3 1 1 1 2
![Page 60: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/60.jpg)
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0 1 0 0 0
3 4 2 2 3 1 1 1 2
![Page 61: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/61.jpg)
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0 1 0 0 0 1
3 4 2 2 3 1 1 1 2
charge(T ) = 3
![Page 62: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/62.jpg)
Hall-Littlewood functions
We have
Hµ[X ; t] =∑T∈Tµ
tch(T )ssh(T )
where Tµ is again the set of all tableaux of evaluation µ. Hence
Hµ[X ; t] =∑S∈Sµ
ε(S)tch(S)gsh(S)
=∑R∈Rµ
tch(R)Gsh(R)
where Sµ (resp. Rµ) is the set of set-valued tableaux (resp. reverseplane partitions) of evaluation µ and the charge of S or R is thecharge of the reading word.
![Page 63: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/63.jpg)
Example
H1,1,1[X ; t] = s1,1,1 + (t + t2)s2,1 + t3s3
=g1,1,1 + (t + t2)g2,1 + t3g3
− 2g1,1 − (t + t2)g2 + g1
321
21 3
31 2 1 2 3
231
312 12 3 1 23 123
![Page 64: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/64.jpg)
Example
H1,1,1[X ; t] = s1,1,1 + (t + t2)s2,1 + t3s3
=g1,1,1 + (t + t2)g2,1 + t3g3
− 2g1,1 − (t + t2)g2 + g1
321
21 3
31 2 1 2 3
231
312 12 3 1 23 123
![Page 65: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/65.jpg)
Littlewood-Richardson tableaux
One version of the Littlewood-Richardson rule states that
sλsµ =∑
T∈Tλ,µ
ssh(T )
where Tλ,µ is the set of tableaux:
I with evaluation (λ1, · · · , λ`(λ), µ1, · · · , µ`(µ))I which have the reverse lattice property with respect to the
letters 1, · · · , `(λ), and
I which have the reverse lattice property with respect to theletters `(λ) + 1, · · · , `(λ) + `(µ)
![Page 66: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/66.jpg)
Littlewood-Richardson tableaux
Example
T(2,1),(2,1) is
2 41 1 3 3
421 1 3 3
2 3 41 1 3
42 31 1 3
32 41 1 3
4321 1 3
3 42 31 1
432 31 1
![Page 67: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/67.jpg)
Littlewood-Richardson tableaux
Corollary
sλsµ =∑
S∈Sλ,µ
ε(S)gsh(S) =∑
R∈Rλ,µ
Gsh(R)
where Sλ,µ (resp. Rλ,µ) is the set of all set-valued tableaux (resp.reverse plane partitions)
I with evaluation (λ1, · · · , λ`(λ), µ1, · · · , µ`(µ))I which have the reverse lattice property with respect to the
letters 1, · · · , `(λ), and
I which have the reverse lattice property with respect to theletters `(λ) + 1, · · · , `(λ) + `(µ)
![Page 68: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/68.jpg)
Grothendieck and Schur functions
Corollary
sλ =∑
S∈Sλ,∅
ε(S)gsh(S)
sλ =∑
R∈Rλ,∅
Gsh(S)
Duality gives expansions of G and g into Schur functions.A different form of these expansions was given by Lenart, in termsof combinatorial objects now called elegant fillings.
![Page 69: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/69.jpg)
Sketch of proof
Given
fα =∑T∈Tα
wt(T )ssh(T )
=∑T∈Tα
wt(T )∑
T ′∈SSYT (sh(T ))
xev(T′)
we want to show
fα =∑S∈Sα
ε(S)wt(S)gsh(S)
=∑S∈Sα
ε(S)wt(S)∑
R∈RPP(sh(S))
xev(R)
![Page 70: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/70.jpg)
Sketch of proof
Given
fα =∑T∈Tα
wt(T )ssh(T )
=∑T∈Tα
wt(T )∑
T ′∈SSYT (sh(T ))
xev(T′)
we want to show
fα =∑S∈Sα
ε(S)wt(S)gsh(S)
=∑S∈Sα
ε(S)wt(S)∑
R∈RPP(sh(S))
xev(R)
![Page 71: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/71.jpg)
The involution
Example
423 41 23 4
21 31 3 4
423 41 23 4
21 31 3 4
![Page 72: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/72.jpg)
The involution
Example
423 41 23 4
21 31 3 4
423 41 23 4
21 31 3 4
![Page 73: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/73.jpg)
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
21 31 3 4
![Page 74: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/74.jpg)
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
•21 31 3 4
![Page 75: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/75.jpg)
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
2•1 31 3 4
![Page 76: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/76.jpg)
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
21• 31 3 4
![Page 77: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/77.jpg)
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
211 31 3 4
![Page 78: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/78.jpg)
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
211 31 3 4
![Page 79: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/79.jpg)
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
21• 31 3 4
![Page 80: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/80.jpg)
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
2•1 31 3 4
![Page 81: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/81.jpg)
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
•21 31 3 4
![Page 82: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/82.jpg)
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
21 31 3 4
![Page 83: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/83.jpg)
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
21 31 3 4
![Page 84: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/84.jpg)
The involution
Example
423 41 23 4
21 31 3 4
423 41 23 4
21 31 3 4
![Page 85: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/85.jpg)
The involution
Example
423 41 23 4
21 31 3 4
423 41 23 4
21 31 3 4
![Page 86: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/86.jpg)
Sage-Combinat meeting tonight
Sage’s mission:
“To create a viable high-quality and open-source alternative toMapleTM, MathematicaTM, MagmaTM, and MATLABTM”
...“and to foster a friendly community of users and developers”
Tonight, Thorton Hall, Room 326
I 7pm-8pm: Introduction to Sage and Sage-Combinat
I 8pm-10pm: Help on installation & getting startedBring your laptop!
I Design discussions
![Page 87: Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof symmetric functions have tableaux{Schur expansions if there exist sets T of semistandard](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0397017e708231d409cd15/html5/thumbnails/87.jpg)
Thank you for your attention.