Drexel University Mathematics Colloquium 2020

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Littlewood–Richardson coefficients and birational combinatorics

Darij Grinberg

14 October 2020

Drexel University, Philadelphia, PA

slides: http:

//www.cip.ifi.lmu.de/~grinberg/algebra/drexel2020.pdf paper: arXiv:2008.06128 aka http:

//www.cip.ifi.lmu.de/~grinberg/algebra/lrhspr.pdf

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Manifest

I shall review the Littlewood–Richardson coefficients and some of their classical properties.

I will then state a “hidden symmetry” conjectured by Pelletier and Ressayre (arXiv:2005.09877) and outline how I proved it.

The proof is a nice example of birational combinatorics: the use of birational transformations in elementary combinatorics (specifically, here, in finding and proving a bijection).

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Manifest

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Manifest

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Chapter 1

Littlewood–Richardson coefficients

References (among many):

Richard Stanley, Enumerative Combinatorics, vol. 2, Chapter 7.

Darij Grinberg, Victor Reiner,Hopf Algebras in Combinatorics, arXiv:1409.8356.

Emmanuel Briand, Mercedes Rosas, The 144 symmetries of the Littlewood-Richardson coefficients of SL3,

arXiv:2004.04995.

Igor Pak, Ernesto Vallejo,Combinatorics and geometry of Littlewood-Richardson cones, arXiv:math/0407170.

Emmanuel Briand, Rosa Orellana, Mercedes Rosas, Rectangular symmetries for coefficients of symmetric functions, arXiv:1410.8017.

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Reminder on symmetric functions

Fix a commutative ring kwith unity. We shall do everything overk.

Consider the ring k[[x₁,x₂,x₃, . . .]] of formal power series in countably many indeterminates.

A formal power series f is said to be bounded-degreeif the monomials it contains are bounded (from above) in degree.

A formal power series f is said to be symmetricif it is invariant under permutations of the indeterminates. For example:

1 +x₁+x₂³ is bounded-degree but not symmetric. (1 +x1) (1 +x2) (1 +x3)· · · is symmetric but not bounded-degree.

Let Λbe the set of all symmetric bounded-degree power series in k[[x1,x2,x3, . . .]]. This is ak-subalgebra, called thering of symmetric functions overk.

It is also known as Sym.

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A formal power series f is said to be symmetricif it is invariant under permutations of the indeterminates.

For example:

1 +x₁+x₂³ is bounded-degree but not symmetric.

(1 +x1) (1 +x2) (1 +x3)· · · is symmetric but not bounded-degree.

Let Λbe the set of all symmetric bounded-degree power series in k[[x1,x2,x3, . . .]]. This is ak-subalgebra, called thering of symmetric functions overk.

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For example:

1 +x₁+x₂³ is bounded-degree but not symmetric.

(1 +x1) (1 +x2) (1 +x3)· · · is symmetric but not bounded-degree.

Let Λ be the set of all symmetric bounded-degree power series in k[[x1,x2,x3, . . .]]. This is ak-subalgebra, called thering of symmetric functions overk.

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Let Λbe the set of all symmetric bounded-degree power series in k[[x₁,x₂,x₃, . . .]]. This is ak-subalgebra, called thering of symmetric functions overk.

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Schur functions, part 1: Young diagrams

Let λ= (λ₁, λ₂, λ₃, . . .) be a partition(i.e., a weakly

decreasing sequence of nonnegative integers such that λi = 0 for all i 0).

We commonly omit trailing zeroes: e.g., the partition

(4,2,2,1,0,0,0,0, . . .) is identified with the tuple (4,2,2,1).

The Young diagramof λis like a matrix, but the rows have different lengths, and are left-aligned; the i-th row hasλ_i cells.

Examples:

The Young diagram of (3,2) has the form .

The Young diagram of (4,2,1) has the form .

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Schur functions, part 1: Young diagrams

Let λ= (λ₁, λ₂, λ₃, . . .) be a partition(i.e., a weakly

decreasing sequence of nonnegative integers such that λi = 0 for all i 0).

We commonly omit trailing zeroes: e.g., the partition

(4,2,2,1,0,0,0,0, . . .) is identified with the tuple (4,2,2,1).

The Young diagramof λis like a matrix, but the rows have different lengths, and are left-aligned; the i-th row hasλ_i cells.

Examples:

The Young diagram of (3,2) has the form .

The Young diagram of (4,2,1) has the form .

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Schur functions, part 2: semistandard tableaux

Asemistandard tableau of shapeλis the Young diagram of λ, filled with positive integers, such that

the entries in eachrow are weaklyincreasing;

the entries in eachcolumn arestrictly increasing.

Examples:

A semistandard tableau of shape (3,2) is

2 3 3

3 5 .

A semistandard tableau of shape (4,2,1) is

2 2 3 4

3 4 5

.

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Schur functions, part 2: semistandard tableaux

Asemistandard tableau of shapeλis the Young diagram of λ, filled with positive integers, such that

the entries in eachrow are weaklyincreasing;

the entries in eachcolumn arestrictly increasing.

Examples:

The semistandard tableaux of shape (3,2) are the arrays of the form

a b c d e

with a≤b≤c andd ≤e and a<d andb <e.

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Schur functions, part 3: definition of Schur functions Given a partitionλ, we define theSchur function s_λ as the power series

s_λ = X

T is a semistandard tableau of shapeλ

x_T, where x_T = Y

pis a cell ofT

x_T(p)

(where T(p) denotes the entry ofT in p).

Examples:

s_(3,2) = X

a≤b≤c,d≤e, a<d,b<e

x_ax_bx_cx_dx_e,

because the semistandard tableau T = a b c

d e

contributes the addend x_T =x_ax_bx_cx_dx_e.

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s_λ = X

x_T, where x_T = Y

pis a cell ofT

x_T(p)

Examples:

For anyn≥0, we have s_(n)= X

i1≤i2≤···≤in

x_i₁x_i₂· · ·x_i_n,

since the semistandard tableaux of shape (n) are the fillings

T = i₁ i₂ · · · i_n with i1≤i2≤ · · · ≤in.

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s_λ = X

x_T, where x_T = Y

pis a cell ofT

x_T(p)

Examples:

For anyn≥0, we have s_(n)= X

i1≤i2≤···≤in

x_i₁x_i₂· · ·x_i_n,

since the semistandard tableaux of shape (n) are the fillings

T = i₁ i₂ · · · i_n with i1≤i2≤ · · · ≤in.

This symmetric function s_(n) is commonly calledh_n.

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s_λ = X

x_T, where x_T = Y

pis a cell ofT

x_T(p)

Examples:

For anyn≥0, consider the partition (1ⁿ) := (1,1, . . . ,1) (withn entries). Then,

s₍₁ⁿ₎ = X

i1<i2<···<in

xi1xi2· · ·xin,

since the semistandard tableaux of shape (1ⁿ) are the fillings T = ⁱ¹

i2 ... ... in

with i₁<i₂ <· · ·<i_n.

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s_λ = X

x_T, where x_T = Y

pis a cell ofT

x_T(p)

Examples:

For anyn≥0, consider the partition (1ⁿ) := (1,1, . . . ,1) (withn entries). Then,

s₍₁ⁿ₎ = X

i1<i2<···<in

xi1xi2· · ·xin,

since the semistandard tableaux of shape (1ⁿ) are the fillings T = ⁱ¹

i2 ... ... in

with i₁<i₂ <· · ·<i_n.

This symmetric function s₍₁ⁿ₎ is commonly called e_n.

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Schur functions, part 4: classical properties

Theorem: The Schur function s_λ is a symmetric function (=

an element of Λ) for any partitionλ.

Theorem: The family (s_λ)_λis a partition is a basis of the k-module Λ.

Theorem: Fixn ≥0. Letλ= (λ1, λ2, . . . , λn) be a partition with at mostn nonzero entries. Then,

sλ(x1,x2, . . . ,xn)

= det

x_i^λ^j^+n−j

1≤i,j≤n

| {z }

this is called analternant

det

x_i^n−j

1≤i,j≤n

| {z }

= Q

1≤i<j≤n(^xi−x_j)

(= the Vandermonde determinant)

.

Here, for any f ∈Λ, we letf (x1,x2, . . . ,xn) denote the result of substituting 0 for x_n+1,x_n+2,x_n+3, . . . in f; this is a symmetricpolynomial in x1,x2, . . . ,xn.

For proofs, see any text on symmetric functions (e.g., Stanley’s EC2, or Grinberg-Reiner, or Mark Wildon’s notes).

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sλ(x1,x2, . . . ,xn)

= det

x_i^λ^j^+n−j

1≤i,j≤n

| {z }

det

x_i^n−j

1≤i,j≤n

| {z }

= Q

.

Here, for any f ∈Λ, we letf (x1,x2, . . . ,xn) denote the result of substituting 0 for x_n+1,x_n+2,x_n+3, . . . in f; this is a

symmetricpolynomial in x1,x2, . . . ,xn.

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sλ(x1,x2, . . . ,xn)

= det

x_i^λ^j^+n−j

1≤i,j≤n

| {z }

det

x_i^n−j

1≤i,j≤n

| {z }

= Q

.

Here, for any f ∈Λ, we letf (x1,x2, . . . ,xn) denote the result of substituting 0 for x_n+1,x_n+2,x_n+3, . . . in f; this is a

symmetricpolynomial in x1,x2, . . . ,xn.

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Littlewood–Richardson coefficients: definition

Ifµ andν are two partitions, thens_µs_ν belongs to Λ (since Λ is a ring) and thus can be written in the form

sµsν = X

λis a partition

c_µ,ν^λ s_λ for somec_µ,ν^λ ∈k (since thes_λ form a basis of Λ).

The coefficients c_µ,ν^λ are integers, and are called the Littlewood–Richardson coefficients.

Example:

Theorem: The coefficients c_µ,ν^λ are nonnegative integers. Various combinatorial interpretations (“Littlewood–Richardson rules”) for them are known.

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sµsν = X

λis a partition

c_µ,ν^λ s_λ

for somec_µ,ν^λ ∈k (since thes_λ form a basis of Λ).

Example:

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sµsν = X

λis a partition

c_µ,ν^λ s_λ

Example:

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sµsν = X

λis a partition

c_µ,ν^λ s_λ

Example:

s_(2,1)s_(3,1)=s_(3,2,1,1)+s_(3,2,2)+s_(3,3,1) +s_(4,1,1,1)+ 2s_(4,2,1)+s_(4,3) +s_(5,1,1)+s_(5,2)

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sµsν = X

λis a partition

c_µ,ν^λ s_λ

Example:

s_(2,1)s_(3,1)=s_(3,2,1,1)+s_(3,2,2)+s_(3,3,1) +s_(4,1,1,1)+2s_(4,2,1)+s_(4,3) +s_(5,1,1)+s_(5,2),

so c(2,1),(3,1)^(4,2,1) = 2 and c(2,1),(3,1)^(3,3,1) = 1.

Theorem: The coefficients c_µ,ν^λ are nonnegative integers.

Various combinatorial interpretations (“Littlewood–Richardson rules”) for them are known.

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sµsν = X

λis a partition

c_µ,ν^λ s_λ

Example:

s_(2,1)s_(3,1)=s_(3,2,1,1)+s_(3,2,2)+s_(3,3,1) +s_(4,1,1,1)+2s_(4,2,1)+s_(4,3) +s_(5,1,1)+s_(5,2),

so c(2,1),(3,1)^(4,2,1) = 2 and c(2,1),(3,1)^(3,3,1) = 1.

Theorem: The coefficients c_µ,ν^λ are nonnegative integers.

Various combinatorial interpretations (“Littlewood–Richardson rules”) for them are known.

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Why Littlewood–Richardson coefficients? 1

Before we say more about Littlewood–Richardson coefficients, let us see where else they appear.

For k=Z, the cohomology ring H^∗(Gr (k,n))

of the complex Grassmannian Gr (k,n) (ofk-subspaces inCⁿ) is isomorphic to

Λ(hn−k+1,hn−k+2, . . . ,hn,ek+1,ek+2,ek+3, . . .)_ideal. The cohomology classes corresponding to the Schur functions s_λ are the Schubert classes– the classes of theSchubert varieties. Roughly speaking, these subdivide Gr (k,n) according to the positions of the pivots in the row-reduced echelon form.

Thus, the Littlewood–Richardson coefficients c_µ,ν^λ are intersection multiplicities of these Schubert varieties. For details, see:

Laurent Manivel,Symmetric Functions, Schubert Polynomials and Degeneracy Loci, AMS/SMF 1998.

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Why Littlewood–Richardson coefficients? 1 For k=Z, the cohomology ring

H^∗(Gr (k,n))

Λ(hn−k+1,hn−k+2, . . . ,h_n,e_k+1,e_k+2,e_k+3, . . .)_ideal. The cohomology classes corresponding to the Schur functions s_λ are the Schubert classes– the classes of theSchubert varieties. Roughly speaking, these subdivide Gr (k,n) according to the positions of the pivots in the row-reduced echelon form.

Thus, the Littlewood–Richardson coefficients c_µ,ν^λ are intersection multiplicities of these Schubert varieties.

For details, see:

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H^∗(Gr (k,n))

For details, see:

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H^∗(Gr (k,n))

For details, see:

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Here is another interpretation of Littlewood–Richardson coefficients, also related to subspaces of a vector space.

Let V be a finite-dimensional vector space.

The Jordan type J(A) of a nilpotent endomorphism

A∈EndV is the partition (λ1, λ2, λ3, . . .) withλi being the size of the i-th largest Jordan block of A.

Pick a nilpotent endomorphismA∈EndV, and let λ=J(λ) be its Jordan type. Let µ andν be two further partitions. When is there an A-invariant vector subspaceW ⊆V with

J(A) =λ, J(A|_W) =µ, J(A/W) =ν? (A/_W is the endomorphism ofV/W induced byA.) Precisely when c_µ,ν^λ 6= 0.

Moreover, the set of all suchW is a subvariety of Gr (k,n), and has c_µ,ν^λ irreducible components.

For details, see:

Marc van Leeuwen,Flag Varieties and Interpretations of Young Tableau Algorithms.

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Why Littlewood–Richardson coefficients? 2 Let V be a finite-dimensional vector space.

The Jordan type J(A)of a nilpotent endomorphism

A∈EndV is the partition (λ₁, λ₂, λ₃, . . .) withλ_i being the size of the i-th largest Jordan block of A.

Pick a nilpotent endomorphismA∈EndV, and let λ=J(λ) be its Jordan type. Let µ andν be two further partitions.

When is there an A-invariant vector subspaceW ⊆V with J(A) =λ, J(A|_W) =µ, J(A/_W) =ν?

(A/_W is the endomorphism ofV/W induced byA.)

Precisely when c_µ,ν^λ 6= 0.

For details, see:

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(A/_W is the endomorphism ofV/W induced byA.) Precisely when c_µ,ν^λ 6= 0.

For details, see:

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For details, see:

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For details, see:

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Fix anN ≥0. The irreducible polynomial representations V_λ of the group GL (N) := GL (N,C) are indexed by partitions having ≤N entries.

Their charactersare the Schur functions s_λ.

The Littlewood–Richardson coefficients tell how to decompose the tensor product of two such representations:

Vµ⊗Vν =M

λ

V^⊕c

λ µ,ν

λ .

For details, see:

William Fulton,Young Tableaux, CUP 1997.

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Vµ⊗Vν =M

λ

V^⊕c

λ µ,ν

λ .

For details, see:

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Vµ⊗Vν =M

λ

V^⊕c

λ µ,ν

λ .

For details, see:

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Skew semistandard tableaux

In order to formulate the classic (or, at least, best known) Littlewood–Richardson rule, we need a

Definition:

Two partitionsλ= (λ₁, λ₂, λ₃, . . .) and

µ= (µ1, µ2, µ3, . . .) are said to satisfyµ⊆λif each i ≥1 satisfies µ_i ≤λ_i.

(Equivalently: if the Young diagram of µis contained in that ofλ.)

Askew partition is a pair (λ, µ) of two partitions satisfying µ⊆λ. Such a pair is denoted byλ/µ.

Ifλ/µis a skew partition, then theYoung diagram of λ/µis obtained from the Young diagram λwhen all cells of the Young diagram ofµare removed.

Semistandard tableaux of shape λ/µare defined just as ones of shape λ, except that we are now only filling the cells of λ/µ.

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Definition:

Semistandard tableaux of shape λ/µare defined just as ones of shape λ, except that we are now only filling the cells of λ/µ.

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Definition:

Example: The Young diagram of (4,2,1)/(1,1) is

Semistandard tableaux of shape λ/µare defined just as ones of shapeλ, except that we are now only filling the cells of λ/µ.

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Definition:

Semistandard tableaux of shape λ/µare defined just as ones of shapeλ, except that we are now only filling the cells of λ/µ.

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Littlewood–Richardson rule: the classical version Littlewood–Richardson rule: Letλ,µand ν be three partitions. Then,c_µ,ν^λ is the number of semistandard tableaux T of shape λ/µsuch that contT =ν and such that

cont (T |_cols≥_j) is a partition for eachj. Here,

contT denotes the sequence (c1,c2,c3, . . .), where ci is the number of entries equal to i in T;

T |_cols≥_j is what obtained from T when the first j −1 columns are deleted.

Example: c(2,1),(3,1)^(4,2,1) = 2 due to the two tableaux 1 1

1 2

and 1 1

2 1

.

The shortest proof is due to Stembridge (using ideas by Gasharov); see John R. Stembridge, A Concise Proof of the Littlewood-Richardson Rule, 2002, or Section 2.6 in

Grinberg-Reiner. 14 / 46

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Littlewood–Richardson rule: the classical version Littlewood–Richardson rule: Letλ,µand ν be three partitions. Then,c_µ,ν^λ is the number of semistandard tableaux T of shape λ/µsuch that contT =ν and such that

cont (T |_cols≥_j) is a partition for eachj. Here,

contT denotes the sequence (c1,c2,c3, . . .), where ci is the number of entries equal to i in T;

T |_cols≥_j is what obtained from T when the first j −1 columns are deleted.

Example: c(2,1),(3,1)^(4,2,1) = 2 due to the two tableaux 1 1

1 2

and 1 1

2 1

.

The shortest proof is due to Stembridge (using ideas by Gasharov); see John R. Stembridge, A Concise Proof of the Littlewood-Richardson Rule, 2002, or Section 2.6 in

Grinberg-Reiner. 14 / 46

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Basic properties of Littlewood–Richardson coefficients Gradedness: c_µ,ν^λ = 0 unless|λ|=|µ|+|ν|, where|κ|

denotes the size(i.e., the sum of the entries) of a partitionκ.

(This is because Λ is a graded ring and the s_λ are homogeneous.)

Transposition symmetry: c_µ,ν^λ =c_µ^λt^t,ν^t, whereκ^t denotes thetranspose of a partitionκ (i.e., the partition whose Young diagram is obtained from that of κby flipping across the main diagonal).

Commutativity: c_µ,ν^λ =c_ν,µ^λ .

(Obvious from the definition, but hard to prove combinatorially using the Littlewood–Richardson rule.)

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Example:

t

=

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Littlewood–Richardson coefficients: more symmetries Fixn ∈N. LetPar[n]be the set of all partitions having at most n nonzero entries.

Ifλ= (λ₁, λ₂, . . . , λ_n)∈Par[n], and ifk ≥0 is such that all entries of λare ≤k, then λ^∨k shall denote the partition

(k−λ_n,k−λn−1, . . . ,k−λ₁)∈Par[n].

This is called thek-complementof λ.

Complementation symmetry I: Let λ, µ, ν∈Par[n] and k ≥0 be such that all entries of λ, µ, ν are≤k. Then,

c_µ,ν^λ =c_ν,µ^λ =c_λ^µ∨k^∨k,ν =c_ν,λ^µ^∨k∨k =c_µ,λ^ν^∨k∨k =c_λ^ν∨k^∨k,µ. Complementation symmetry II: Let λ, µ, ν∈Par[n] and q,r ≥0 be such that all entries of µare ≤q, and all entries of ν are≤r. Then:

If all entries of λare ≤q+r, then c_µ,ν^λ =c_µ^λ∨q^∨(q+r),ν^∨r. If not, thenc_µ,ν^λ = 0.

(See, e.g., Exercise 2.9.16 in Grinberg-Reiner.)

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(k−λ_n,k−λn−1, . . . ,k−λ₁)∈Par[n].

Example: Ifn= 5, then

(3,1,1)^∨7= (3,1,1,0,0)^∨7= (7−0,7−0,7−1,7−1,7−3)

= (7,7,6,6,4).

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(k−λ_n,k−λn−1, . . . ,k−λ₁)∈Par[n].

Illustration: Ifn= 3, then (3,2)^∨4 =

∨4

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(k−λ_n,k−λn−1, . . . ,k−λ₁)∈Par[n].

Illustration: Ifn= 3, then (3,2)^∨4 =

∨4

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(k−λ_n,k−λn−1, . . . ,k−λ₁)∈Par[n].

Illustration: Ifn= 3, then (3,2)^∨4=

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(k−λ_n,k−λn−1, . . . ,k−λ₁)∈Par[n].

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(k−λ_n,k−λn−1, . . . ,k−λ₁)∈Par[n].

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(k−λ_n,k−λn−1, . . . ,k−λ₁)∈Par[n].

Illustration: Ifn= 3, then

(3,2)^∨4 = (4,2,1).

c_µ,ν^λ =c_ν,µ^λ =c_λ^µ∨k^∨k,ν =c_ν,λ^µ^∨k∨k =c_µ,λ^ν^∨k∨k =c_λ^ν∨k^∨k,µ.

Complementation symmetry II: Let λ, µ, ν∈Par[n] and q,r ≥0 be such that all entries of µare ≤q, and all entries of ν are≤r. Then:

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(k−λ_n,k−λn−1, . . . ,k−λ₁)∈Par[n].

c_µ,ν^λ =c_ν,µ^λ =c_λ^µ∨k^∨k,ν =c_ν,λ^µ^∨k∨k =c_µ,λ^ν^∨k∨k =c_λ^ν∨k^∨k,µ.

Complementation symmetry II: Let λ, µ, ν∈Par[n] and q,r ≥0 be such that all entries of µare ≤q, and all entries of ν are≤r. Then:

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(k−λ_n,k−λn−1, . . . ,k−λ₁)∈Par[n].

c_µ,ν^λ =c_ν,µ^λ =c_λ^µ∨k^∨k,ν =c_ν,λ^µ^∨k∨k =c_µ,λ^ν^∨k∨k =c_λ^ν∨k^∨k,µ. (This can be proved by applying skew Schur functions to x₁⁻¹,x₂⁻¹, . . . ,x_n⁻¹, or by interpreting Schur functions as fundamental classes in the cohomology of the Grassmannian.

See Exercise 2.9.15 in Grinberg-Reiner for the former proof.) Complementation symmetry II: Let λ, µ, ν∈Par[n] and q,r ≥0 be such that all entries of µare ≤q, and all entries of ν are ≤r. Then:

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(k−λ_n,k−λn−1, . . . ,k−λ₁)∈Par[n].

c_µ,ν^λ =c_ν,µ^λ =c_λ^µ∨k^∨k,ν =c_ν,λ^µ^∨k∨k =c_µ,λ^ν^∨k∨k =c_λ^ν∨k^∨k,µ. Complementation symmetry II: Let λ, µ, ν∈Par[n] and q,r ≥0 be such that all entries of µare ≤q, and all entries of ν are ≤r. Then:

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The Briand–Rosas symmetry

In arXiv:2004.04995, Emmanuel Briand and Mercedas Rosas have used a computer (and prior work of Rassart, Knutson and Tao, which made the problem computable) to classify all such “symmetries” of Littlewood–Richardson coefficientsc_µ,ν^λ with λ, µ, ν∈Par[n] for fixed

n ∈ {3,4, . . . ,7}.

For n∈ {4,5, . . . ,7}, they only found the complementation symmetries above, as well as the trivial translation symmetries (adding 1 to each entry of λandν does not changec_µ,ν^λ ; nor does adding 1 to each entry ofλand µ).

For n= 3, they found an extra symmetry: c_(µ^(λ¹^,λ²^,λ³⁾

1,µ2),(ν1,ν2)=c_(µ^(λ¹^,ν¹^,λ³⁾

1+ν1−λ2,µ2+ν1−λ2),(λ2,ν2) .

(Read the right hand side as 0 if the tuples are not partitions.) Question: Is there a non-computer proof? What is the meaning of this identity?

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