nilpotent ideals of upper triangular matrices and ...nilpotent ideals of upper triangular matrices...

Nilpotent ideals of upper triangular matrices andVariations of the Catalan numbers

Jia Huang

University of Nebraska at KearneyE-mail address: [email protected]

October 19, 2017

Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 1 / 25

Ideals of upper triangular matrices

Definition

Let Un be the algebra of all n-by-n upper triangular matrices∗ ∗ ∗ · · · ∗0 ∗ ∗ · · · ∗0 0 ∗ · · · ∗...

.... . .

. . ....

0 0 · · · 0 ∗

where a star ∗ is an arbitrary entry from a fixed field F (e.g., R).

A (two-sided) ideal I of Un is a vector subspace of Un such thatXI ⊆ I and IX ⊆ I for all X ∈ Un.

A ideal I is nilpotent if I k = 0 for some k ≥ 1. The smallest k suchthat I k = 0 is the (nilpotent) order of I .

A ideal I of Un is commutative if AB = BA for all A,B ∈ I .

Definition

.... . .

. . ....

0 0 · · · 0 ∗

Definition

.... . .

. . ....

0 0 · · · 0 ∗

Definition

.... . .

. . ....

0 0 · · · 0 ∗

Nilpotent ideals

Example (A nilpotent ideal of U6 and its corresponding Dyck path)

0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0

height = 3

Observation

An nilpotent ideal of Un is represented by a matrix of 0’s and ∗’sseparated by a Dyck path of length 2n.

The number of such ideals is the Catalan number Cn := 1n+1

The number of all ideals of Un is the Catalan number Cn+1.

Nilpotent ideals

0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0

height = 3

Observation

Nilpotent ideals

0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0

height = 3

Observation

Nilpotent ideals

0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0

height = 3

Observation

Nilpotent ideals

0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0

height = 3

Observation

Commutative ideals

Proposition (L. Shapiro, 1975)

The number of commutative ideals of Un is 2n−1.

Problem

Find a direct proof of the above result.

Example

The number of subsets of {1, 2, . . . , n} is(n0

)+ · · ·+

)= 2n.

This can be proved by considering whether a subset contains i for each i .

Observation

An ideal of Un is commutative if and only if it has nilpotent order 1 or 2.

Definition

Let Cdn be the number of nilpotent ideals of Un with order at most d .

Commutative ideals

Problem

Example

)+ · · ·+

)= 2n.

Observation

Definition

Commutative ideals

Problem

Example

)+ · · ·+

)= 2n.

Observation

Definition

Commutative ideals

Problem

Example

)+ · · ·+

)= 2n.

Observation

Definition

Commutative ideals

Problem

Example

)+ · · ·+

)= 2n.

Observation

Definition

Nilpotent order

Observation

The order of a nilpotent ideal I of Un is the largest possible length d of anadmissible sequence, that is, a sequence (i1, i2, . . . , id) such that the entry(ij , ij+1) is a star ∗ in the matrix form of I for all j = 1, 2, . . . , d − 1.

Example

The following ideal has nilpotent order is 4 since the sequence (1, 3, 5, 6) isadmissible and there is no longer admissible sequence.

0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0

Nilpotent order

Observation

The order of a nilpotent ideal I of Un is the largest possible length d of anadmissible sequence, that is, a sequence (i1, i2, . . . , id) such that the entry(ij , ij+1) is a star ∗ in the matrix form of I for all j = 1, 2, . . . , d − 1.

Example

The following ideal has nilpotent order is 4 since the sequence (1, 3, 5, 6) isadmissible and there is no longer admissible sequence.

0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0

Bounce Paths

Observation

Let I be an ideal of Un corresponding to a Dyck path D. Then thenilpotent order of I is the number of times the bounce path of D bouncesoff the main diagonal.

Example (Bounce Path)

The bounce path has 4 bounces.

The Dyck path D has height 3.

Fact (Andrews–Krattenthaler–Orsina–Papi 2002, Haglund 2008)

Bijection ζ : Dyck paths with height d ↔ Dyck paths with d bounces.

Bounce Paths

Observation

Bounce Paths

Observation

Bounce Paths

Observation

Generalization of Commutative Ideals

Theorem (H.-Rhoades)

Dyck paths of length 2n with height at most d are counted by Cdn . Hence

Cdn is the sequence A080934 in OEIS and interpolates between 1 and Cn.

Example

n 1 2 3 4 5 6 7 n

C 1n 1 1 1 1 1 1 1 1

C 2n 1 2 4 8 16 32 64 2n−1

C 3n 1 2 5 13 34 89 233 F2n−1

C 4n 1 2 5 14 41 122 365 1

2(1 + 3n−1)

Cn 1 2 5 14 42 132 429 1n+1

)Problem

Is there a nice (q, t)-analogue of the number Cdn ?

Example

n 1 2 3 4 5 6 7 n

C 1n 1 1 1 1 1 1 1 1

C 2n 1 2 4 8 16 32 64 2n−1

C 3n 1 2 5 13 34 89 233 F2n−1

C 4n 1 2 5 14 41 122 365 1

2(1 + 3n−1)

Cn 1 2 5 14 42 132 429 1n+1

Problem

Example

n 1 2 3 4 5 6 7 n

C 1n 1 1 1 1 1 1 1 1

C 2n 1 2 4 8 16 32 64 2n−1

C 3n 1 2 5 13 34 89 233 F2n−1

C 4n 1 2 5 14 41 122 365 1

2(1 + 3n−1)

Cn 1 2 5 14 42 132 429 1n+1

)Problem

Ideals of Lie Algebras

Definition

Let sln(C) be the (type A semisimple) Lie algebra of all n× n complexmatrices with zero trace under the Lie bracket [X ,Y ] := XY − YX .

Let b be the Borel subalgebra of upper triangular matrices of sln(C).

Theorem (Andrews–Krattenthaler–Orsina–Papi 2002)

The number of ad-nilpotent ideals of b with order at most d − 1 is Cdn .

This gives a nonsymmetric (q, t)-analogue of Cn using (#bounces, area).

Problem

Find a natural order-preserving bijection between nilpotent ideals ofUn and ad-nilpotent ideals of b. (The exponential map?)

The above theorem has been generalized from type A to other types[Krattenthaler–Orsina–Papi 2002]. Is there a similar generalization fornilpotent ideals of Un?

Definition

Problem

Definition

Problem

Definition

Problem

Definition

Problem

Generating function

Definition

Let Cd(x) :=∑

n≥0 Cdn x

n+1 for d ≥ 1, and let C 0(x) := x .

Let Fi (x) := i for i = 0, 1, and Fn(x) := Fn−1(x)− xFn−2(x), n ≥ 2.

Proposition (de Bruijn–Knuth–Rice 1972)

For n ≥ 1 we have Fn(x) =∑

0≤i≤(n−1)/2(n−1−i

)(−x)i .

Proposition (Kreweras 1970)

For d ≥ 1 we have Cd(x) =x

1− Cd−1(x)=

xFd+1(x)

Fd+2(x).

Example

C 1(x) = x1−x , C 2(x) = x

1− x1−x

= x(1−x)1−2x , C 3(x) = x

1− x1− x

= x(1−2x)1−3x+x2

Generating function

Definition

Let Cd(x) :=∑

n≥0 Cdn x

n+1 for d ≥ 1, and let C 0(x) := x .

0≤i≤(n−1)/2(n−1−i

)(−x)i .

1− Cd−1(x)=

xFd+1(x)

Fd+2(x).

Example

C 1(x) = x1−x , C 2(x) = x

1− x1−x

= x(1−x)1−2x , C 3(x) = x

1− x1− x

= x(1−2x)1−3x+x2

Generating function

Definition

Let Cd(x) :=∑

n≥0 Cdn x

n+1 for d ≥ 1, and let C 0(x) := x .

0≤i≤(n−1)/2(n−1−i

)(−x)i .

1− Cd−1(x)=

xFd+1(x)

Fd+2(x).

Example

C 1(x) = x1−x , C 2(x) = x

1− x1−x

= x(1−x)1−2x , C 3(x) = x

1− x1− x

= x(1−2x)1−3x+x2

Generating function

Definition

Let Cd(x) :=∑

n≥0 Cdn x

n+1 for d ≥ 1, and let C 0(x) := x .

0≤i≤(n−1)/2(n−1−i

)(−x)i .

1− Cd−1(x)=

xFd+1(x)

Fd+2(x).

Example

C 1(x) = x1−x , C 2(x) = x

1− x1−x

= x(1−x)1−2x , C 3(x) = x

1− x1− x

= x(1−2x)1−3x+x2

Generating function

Definition

Let Cd(x) :=∑

n≥0 Cdn x

n+1 for d ≥ 1, and let C 0(x) := x .

0≤i≤(n−1)/2(n−1−i

)(−x)i .

1− Cd−1(x)=

xFd+1(x)

Fd+2(x).

Example

C 1(x) = x1−x , C 2(x) = x

1− x1−x

= x(1−x)1−2x , C 3(x) = x

1− x1− x

= x(1−2x)1−3x+x2

Closed Formulas for C dn

The number Cdn has the following closed formulas:

∑i∈Z

2i(d + 2) + 1

2n + 1

(2n + 1

n − i(d + 2)

)= det

[(i − j + d

j − i + 1

)]n−1i ,j=1

0=i0≤i1≤···≤id−1≤id=n

∏0≤j≤d−2

(ij+2 − ij − 1

ij+1 − ij

Theorem (de Bruijn–Knuth–Rice 1972)

The number of plane trees with n + 1 nodes of depth at most d is

∑1≤j≤d+1

sin2(jπ/(d + 2)) cos2n(jπ/(d + 2)).

∑i∈Z

2i(d + 2) + 1

2n + 1

(2n + 1

n − i(d + 2)

)= det

[(i − j + d

j − i + 1

)]n−1i ,j=1

0=i0≤i1≤···≤id−1≤id=n

∏0≤j≤d−2

(ij+2 − ij − 1

ij+1 − ij

∑1≤j≤d+1

sin2(jπ/(d + 2)) cos2n(jπ/(d + 2)).

Nonassociativity of binary operations

Let ∗ be a binary operation on a set X , and let x0, x1, . . . , xn beX -valued indeterminates.

There are Cn ways to parenthesize x0 ∗ x1 ∗ · · · ∗ xn.

The number C∗,n of distinct results from parenthesizations ofx0 ∗ x1 ∗ · · · ∗ xn satisfies 1 ≤ C∗,n ≤ Cn.

We have C∗,n = 1 for all n ≥ 0 if and only if ∗ is associative. So ingeneral, C∗,n measures the failure of ∗ to be associative.

Example (Subtraction, n = 3)

((x0−x1)−x2)−x3

= x0 − x1 − x2 − x3

(x0−(x1−x2))−x3

= x0 − x1 + x2 − x3

(x0−x1)−(x2−x3)

= x0 − x1 − x2 + x3

x0−((x1−x2)−x3)

= x0 − x1 + x2 + x3

x0−(x1−(x2−x3))

= x0 − x1 + x2 − x3

C3 = 5

C−,3 = 4

((x0−x1)−x2)−x3

= x0 − x1 − x2 − x3

(x0−(x1−x2))−x3

= x0 − x1 + x2 − x3

(x0−x1)−(x2−x3)

= x0 − x1 − x2 + x3

x0−((x1−x2)−x3)

= x0 − x1 + x2 + x3

x0−(x1−(x2−x3))

= x0 − x1 + x2 − x3

C3 = 5

C−,3 = 4

((x0−x1)−x2)−x3

= x0 − x1 − x2 − x3

(x0−(x1−x2))−x3

= x0 − x1 + x2 − x3

(x0−x1)−(x2−x3)

= x0 − x1 − x2 + x3

x0−((x1−x2)−x3)

= x0 − x1 + x2 + x3

x0−(x1−(x2−x3))

= x0 − x1 + x2 − x3

C3 = 5

C−,3 = 4

((x0−x1)−x2)−x3

= x0 − x1 − x2 − x3

(x0−(x1−x2))−x3

= x0 − x1 + x2 − x3

(x0−x1)−(x2−x3)

= x0 − x1 − x2 + x3

x0−((x1−x2)−x3)

= x0 − x1 + x2 + x3

x0−(x1−(x2−x3))

= x0 − x1 + x2 − x3

C3 = 5

C−,3 = 4

((x0−x1)−x2)−x3

= x0 − x1 − x2 − x3

(x0−(x1−x2))−x3

= x0 − x1 + x2 − x3

(x0−x1)−(x2−x3)

= x0 − x1 − x2 + x3

x0−((x1−x2)−x3)

= x0 − x1 + x2 + x3

x0−(x1−(x2−x3))

= x0 − x1 + x2 − x3

C3 = 5

C−,3 = 4

((x0−x1)−x2)−x3 = x0 − x1 − x2 − x3(x0−(x1−x2))−x3 = x0 − x1 + x2 − x3(x0−x1)−(x2−x3) = x0 − x1 − x2 + x3x0−((x1−x2)−x3) = x0 − x1 + x2 + x3x0−(x1−(x2−x3)) = x0 − x1 + x2 − x3

C3 = 5

C−,3 = 4

((x0−x1)−x2)−x3 = x0 − x1 − x2 − x3(x0−(x1−x2))−x3 = x0 − x1 + x2 − x3(x0−x1)−(x2−x3) = x0 − x1 − x2 + x3x0−((x1−x2)−x3) = x0 − x1 + x2 + x3x0−(x1−(x2−x3)) = x0 − x1 + x2 − x3

C3 = 5

C−,3 = 4

A four-parameter generalization of C dn

Definition

Define a binary operation ⊕ on R := C[x , y ]/(xd+k − xd , y e+` − y e):

f ⊕ g := xf + yg ∀f , g ∈ R.

Let Cd ,ek,`,n := C⊕,n be the number of distinct results obtained by

inserting parentheses to the expression z0 ⊕ z1 ⊕ · · · ⊕ zn, wherez0, z1, . . . , zn are indeterminates taking values in R.

Example (n = 3, d = 2, e = k = ` = 1)

For R := C[x , y ]/(x2+1 − x2, y1+1 − y1) we have

((z0 ⊕ z1)⊕ z2)⊕ z3

= x3z0 + x2yz1 + xyz2 + yz3

(z0 ⊕ (z1 ⊕ z2))⊕ z3

= x2z0 + x2yz1 + xy2z2 + yz3

(z0 ⊕ z1)⊕ (z2 ⊕ z3)

= x2z0 + xyz1 + xyz2 + y2z3

z0 ⊕ ((z1 ⊕ z2)⊕ z3)

= xz0 + x2yz1 + xy2z2 + y2z3

z0 ⊕ (z1 ⊕ (z2 ⊕ z3))

= xz0 + xyz1 + xy2z2 + y3z3

⇒C5 = 5 = 1

)C 2,11,1,3 = 4 = 2n−1

Definition

f ⊕ g := xf + yg ∀f , g ∈ R.

Example (n = 3, d = 2, e = k = ` = 1)

((z0 ⊕ z1)⊕ z2)⊕ z3

= x3z0 + x2yz1 + xyz2 + yz3

(z0 ⊕ (z1 ⊕ z2))⊕ z3

= x2z0 + x2yz1 + xy2z2 + yz3

(z0 ⊕ z1)⊕ (z2 ⊕ z3)

= x2z0 + xyz1 + xyz2 + y2z3

z0 ⊕ ((z1 ⊕ z2)⊕ z3)

= xz0 + x2yz1 + xy2z2 + y2z3

z0 ⊕ (z1 ⊕ (z2 ⊕ z3))

= xz0 + xyz1 + xy2z2 + y3z3

⇒C5 = 5 = 1

)C 2,11,1,3 = 4 = 2n−1

Definition

f ⊕ g := xf + yg ∀f , g ∈ R.

Example (n = 3, d = 2, e = k = ` = 1)

((z0 ⊕ z1)⊕ z2)⊕ z3

= x3z0 + x2yz1 + xyz2 + yz3

(z0 ⊕ (z1 ⊕ z2))⊕ z3

= x2z0 + x2yz1 + xy2z2 + yz3

(z0 ⊕ z1)⊕ (z2 ⊕ z3)

= x2z0 + xyz1 + xyz2 + y2z3

z0 ⊕ ((z1 ⊕ z2)⊕ z3)

= xz0 + x2yz1 + xy2z2 + y2z3

z0 ⊕ (z1 ⊕ (z2 ⊕ z3))

= xz0 + xyz1 + xy2z2 + y3z3

⇒C5 = 5 = 1

)C 2,11,1,3 = 4 = 2n−1

Definition

f ⊕ g := xf + yg ∀f , g ∈ R.

Example (n = 3, d = 2, e = k = ` = 1)

((z0 ⊕ z1)⊕ z2)⊕ z3 = x3z0 + x2yz1 + xyz2 + yz3(z0 ⊕ (z1 ⊕ z2))⊕ z3 = x2z0 + x2yz1 + xy2z2 + yz3(z0 ⊕ z1)⊕ (z2 ⊕ z3) = x2z0 + xyz1 + xyz2 + y2z3z0 ⊕ ((z1 ⊕ z2)⊕ z3) = xz0 + x2yz1 + xy2z2 + y2z3z0 ⊕ (z1 ⊕ (z2 ⊕ z3)) = xz0 + xyz1 + xy2z2 + y3z3

⇒C5 = 5 = 1

)C 2,11,1,3 = 4 = 2n−1

Definition

f ⊕ g := xf + yg ∀f , g ∈ R.

Example (n = 3, d = 2, e = k = ` = 1)

((z0 ⊕ z1)⊕ z2)⊕ z3 = x3z0 + x2yz1 + xyz2 + yz3(z0 ⊕ (z1 ⊕ z2))⊕ z3 = x2z0 + x2yz1 + xy2z2 + yz3(z0 ⊕ z1)⊕ (z2 ⊕ z3) = x2z0 + xyz1 + xyz2 + y2z3z0 ⊕ ((z1 ⊕ z2)⊕ z3) = xz0 + x2yz1 + xy2z2 + y2z3z0 ⊕ (z1 ⊕ (z2 ⊕ z3)) = xz0 + xyz1 + xy2z2 + y3z3

⇒C5 = 5 = 1

)C 2,11,1,3 = 4 = 2n−1

Binary trees

Parenthesizations of z0 ⊕ · · · ⊕ zn ↔ (full) binary trees with n + 1 leaves.

Example

3 0 1 2 3 0

l l l l l((z0⊕z1)⊕z2)⊕z3 (z0⊕(z1⊕z2))⊕z3 (z0⊕z1)⊕(z2⊕z3) z0⊕((z1⊕z2)⊕z3) z0⊕(z1⊕(z2⊕z3))

δ = (3, 2, 1, 0) δ = (2, 2, 1, 0) δ = (2, 1, 1, 0) δ = (1, 2, 1, 0) δ = (1, 1, 1, 0)ρ = (0, 1, 1, 1) ρ = (0, 1, 2, 1) ρ = (0, 1, 1, 2) ρ = (0, 1, 2, 2) ρ = (0, 1, 2, 3)

Observation

A parenthesization of z0 ⊕ · · · ⊕ zn corresponding to t ∈ Tn equals

xδ0(t)yρ0(t)z0 + · · ·+ xδn(t)yρn(t)znwhere the left depth δi (t) (or right depth ρi (t), resp.) of leaf i in t ∈ Tn isthe number of edges to the left (or right, resp.) in the unique path fromthe root of t down to i .

Binary trees

Example

3 0 1 2 3 0

δ = (3, 2, 1, 0) δ = (2, 2, 1, 0) δ = (2, 1, 1, 0) δ = (1, 2, 1, 0) δ = (1, 1, 1, 0)ρ = (0, 1, 1, 1) ρ = (0, 1, 2, 1) ρ = (0, 1, 1, 2) ρ = (0, 1, 2, 2) ρ = (0, 1, 2, 3)

Observation

Binary trees

Example

3 0 1 2 3 0

δ = (3, 2, 1, 0) δ = (2, 2, 1, 0) δ = (2, 1, 1, 0) δ = (1, 2, 1, 0) δ = (1, 1, 1, 0)ρ = (0, 1, 1, 1) ρ = (0, 1, 2, 1) ρ = (0, 1, 1, 2) ρ = (0, 1, 2, 2) ρ = (0, 1, 2, 3)

Observation

Associativity at left depth d

Definition (Associativity and Rotation)

A binary operation ∗ is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) alwaysholds.

The rotation for binary trees is given byr s t

↔r s t

Observation

If e = k = ` = 1 then two parenthesizations of z0 ⊕ · · · ⊕ zn are equal ifand only if their corresponding binary trees can be obtained from eachother by applying rotations at nodes of left depth at least d − 1.

Example

The first two binary trees are equivalent for ⊕ with d = 2, e = k = ` = 1.

3 0 1 2 3 0

↔r s t

Observation

Example

3 0 1 2 3 0

↔r s t

Observation

Example

3 0 1 2 3 0

↔r s t

Observation

Example

3 0 1 2 3 0

nilpotent ideals of upper triangular matrices and ...nilpotent ideals of upper triangular matrices...

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