The Combinatorics of Young Tableaux

The combinatorics of Young tableaux

John A. Miller

Young Diagrams The combinatorics of Young tableaux Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux John A. Miller Counting tableaux

SST as a monoid Baylor University Skew tableaux Jeu de taquin The product on SST

Colloquium Factoring Young tableaux Sam Houston State University Littlewood-Richardson numbers April 10, 2019 The Pieri rule The Littlewood-Richardson rule

e-mail: [email protected] The combinatorics Young Tableaux: Endgame of Young tableaux John A. Miller

Our research: Young Diagrams ▶ Partitions syzygies of modules of covariants Young diagrams The poset L(m, n) ▶ minimal free resolutions in classical invariant theory Young tableaux ▶ Young tableaux parabolic BGG category O Counting tableaux SST as a monoid Skew tableaux Proofs (for the most part) come down combinatorics of Jeu de taquin Young tableaux. The product on SST Factoring Young tableaux Littlewood-Richardson numbers What I want you to get out of today’s talk The Pieri rule The Littlewood-Richardson rule ▶ A flavor of the combinatorics of Young tableaux ▶ A sense of how Young tableaux can be applied The combinatorics TOC of Young tableaux John A. Miller

Young Diagrams Young Diagrams Partitions Partitions Young diagrams Young diagrams The poset L(m, n) Young tableaux The poset L(m, n) Young tableaux Counting tableaux

Young tableaux SST as a monoid Young tableaux Skew tableaux Jeu de taquin Counting tableaux The product on SST Factoring Young SST as a monoid tableaux Littlewood-Richardson Skew tableaux numbers The Pieri rule The Littlewood-Richardson Jeu de taquin rule The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics of Young tableaux

John A. Miller

Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

Young Diagrams SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Alfred Young, 1873 - 1940 of Young tableaux John A. Miller

Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young ▶ tableaux Born Widnes, Lancashire, England 1873 Littlewood-Richardson numbers ▶ Introduced Young tableaux in 1900 while working in The Pieri rule The Littlewood-Richardson classical invariant theory rule ▶ Work extended by Frobenius, Schur, Weyl in early 1900s ▶ Only finitely many λi ≠ 0 ▶ Each λi ≠ 0 is a part of λ ▶ λ ⊢ n or |λ| = n

λ = (4, 2, 1, 0, 0,...) = (4, 2, 1, 0, 0) = (4, 2, 1) |λ| = 7

The combinatorics Integer Partitions of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A partition of n ∈ N is a sequence of integers The poset L(m, n) Young tableaux Young tableaux λ = (λ1, λ2,...) Counting tableaux

∑ SST as a monoid where λ1 ≥ λ2 ≥ · · · ≥ 0 and λi = n. Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule λ = (4, 2, 1, 0, 0,...) = (4, 2, 1, 0, 0) = (4, 2, 1) |λ| = 7

The combinatorics Integer Partitions of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A partition of n ∈ N is a sequence of integers The poset L(m, n) Young tableaux Young tableaux λ = (λ1, λ2,...) Counting tableaux

∑ SST as a monoid where λ1 ≥ λ2 ≥ · · · ≥ 0 and λi = n. Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux ▶ Littlewood-Richardson Only finitely many λi ≠ 0 numbers The Pieri rule ▶ ̸ The Littlewood-Richardson Each λi = 0 is a part of λ rule ▶ λ ⊢ n or |λ| = n The combinatorics Integer Partitions of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A partition of n ∈ N is a sequence of integers The poset L(m, n) Young tableaux Young tableaux λ = (λ1, λ2,...) Counting tableaux

∑ SST as a monoid where λ1 ≥ λ2 ≥ · · · ≥ 0 and λi = n. Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux ▶ Littlewood-Richardson Only finitely many λi ≠ 0 numbers The Pieri rule ▶ ̸ The Littlewood-Richardson Each λi = 0 is a part of λ rule ▶ λ ⊢ n or |λ| = n

λ = (4, 2, 1, 0, 0,...) = (4, 2, 1, 0, 0) = (4, 2, 1) |λ| = 7 The combinatorics Partitions of Young tableaux John A. Miller

There are eleven partitions of 6: Young Diagrams Partitions Young diagrams (6), (5, 1), (4, 2)(4, 1, 1) The poset L(m, n) Young tableaux Young tableaux (3, 3), (3, 2, 1), (3, 1, 1, 1), (2, 2, 2) Counting tableaux

SST as a monoid (2, 2, 1, 1), (2, 1, 1, 1, 1), (1, 1, 1, 1, 1, 1) Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux There is only one partition of zero, the empty partition Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson ∅ = (0) rule Examples:

λ = (4, 2, 1) ←→ λ =

σ = (6, 6, 3, 3, 1, 1) ←→ σ =

The combinatorics Partitions and Young diagrams of Young tableaux John A. Miller

Partitions of n can be represented by a Young diagram of Young Diagrams Partitions size n. Young diagrams The poset L(m, n)

A Young diagram of size n is an array of n left-justified Young tableaux boxes with weakly decreasing row length. Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Partitions and Young diagrams of Young tableaux John A. Miller

Partitions of n can be represented by a Young diagram of Young Diagrams Partitions size n. Young diagrams The poset L(m, n)

A Young diagram of size n is an array of n left-justified Young tableaux boxes with weakly decreasing row length. Young tableaux Counting tableaux Examples: SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young λ = (4, 2, 1) ←→ λ = tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

σ = (6, 6, 3, 3, 1, 1) ←→ σ = The combinatorics Partitions and Young diagrams of Young tableaux John A. Miller

More examples: ∅ ←→ · Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux ←→ Young tableaux (3, 3, 2, 2) Counting tableaux

(1, 1, 1, 1, 1, 1, 1) ←→ SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers (5, 4, 3, 2, 1) ←→ The Pieri rule The Littlewood-Richardson rule

(7) ←→ The combinatorics Partial ordering on Young diagrams of Young tableaux John A. Miller

For two Young diagrams λ and σ, we write Young Diagrams Partitions Young diagrams λ ⊂ σ The poset L(m, n) Young tableaux Young tableaux if σ contains the diagram λ as a subset. Counting tableaux

SST as a monoid Skew tableaux If λ = (λ1, λ2,...) and σ = (σ1, σ2,...), this is equivalent to Jeu de taquin The product on SST λ ≤ σ for all i. Factoring Young i i tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule ⊂ i.e. L(m, n) = all Young diagrams with ≤ m rows and ≤ n columns ⊂ ··· i.e. L(m, n) = all Young diagrams (|n, {z , n}) m

The combinatorics The poset L(m, n) of Young tableaux John A. Miller

L(m, n) = all Young diagrams that fit in an m × n rectangle, Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule i.e. L(m, n) = all Young diagrams with ≤ m rows and ≤ n columns ⊂ ··· i.e. L(m, n) = all Young diagrams (|n, {z , n}) m

The combinatorics The poset L(m, n) of Young tableaux John A. Miller

L(m, n) = all Young diagrams that fit in an m × n rectangle, Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin e.g. L(4, 6), The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule i.e. L(m, n) = all Young diagrams with ≤ m rows and ≤ n columns ⊂ ··· i.e. L(m, n) = all Young diagrams (|n, {z , n}) m

The combinatorics The poset L(m, n) of Young tableaux John A. Miller

L(m, n) = all Young diagrams that fit in an m × n rectangle, Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin e.g. L(4, 6), λ = (5, 3, 1), The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule ⊂ ··· i.e. L(m, n) = all Young diagrams (|n, {z , n}) m

The combinatorics The poset L(m, n) of Young tableaux John A. Miller

L(m, n) = all Young diagrams that fit in an m × n rectangle, Young Diagrams Partitions i.e. L(m, n) = all Young diagrams with ≤ m rows and ≤ n Young diagrams The poset L(m, n)

columns Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin e.g. L(4, 6), λ = (5, 3, 1), The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics The poset L(m, n) of Young tableaux John A. Miller

L(m, n) = all Young diagrams that fit in an m × n rectangle, Young Diagrams Partitions i.e. L(m, n) = all Young diagrams with ≤ m rows and ≤ n Young diagrams The poset L(m, n)

columns Young tableaux Young tableaux ⊂ ··· Counting tableaux i.e. L(m, n) = all Young diagrams (n|, {z , n}) SST as a monoid m Skew tableaux Jeu de taquin e.g. L(4, 6), λ = (5, 3, 1), The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

U The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

UR The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

URU The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

URUR The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

URURR The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

URURRU The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

URURRUR The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

URURRURR The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

URURRURRU The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

URURRURRUR The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

URURRURRUR λ corresponds to a choice of 4 U’s (equiv. 6 R’s) in 10 steps The combinatorics The size of L(m, n) of Young tableaux ( ) ( ) John A. Miller m + n (m + n)! m + n L(m, n) has size = = Young Diagrams n n!m! m Partitions Young diagrams The poset L(m, n) Idea: each Young diagram λ ∈ L(m, n) can be identified with Young tableaux Young tableaux a sequence of up steps U and right steps R: Counting tableaux

SST as a monoid | | Skew tableaux e.g. L(4, 6), λ = (5, 3, 1), so L(4, 6) = 210 Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

URURRURRUR λ corresponds to a choice of 4 U’s (equiv. 6 R’s) in 10 steps The combinatorics Partial ordering on L(m, n) of Young tableaux John A. Miller

e.g. L(2, 2) Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Partial ordering on L(m, n) of Young tableaux John A. Miller

e.g. L(2, 2) Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux · SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Partial ordering on L(m, n) of Young tableaux John A. Miller

e.g. L(2, 2) Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux · SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

Our work - these posets parametrize minimal free resolutions for modules of covariants. The combinatorics Partial ordering on L(m, n) of Young tableaux John A. Miller

e.g. L(2, 3) Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Partial ordering on L(m, n) of Young tableaux John A. Miller

e.g. L(2, 3) Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid · Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Partial ordering on L(m, n) of Young tableaux John A. Miller

e.g. L(2, 3) Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid · Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics of Young tableaux

John A. Miller

Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

Young tableaux SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule Examples:

1 10 2 1 4 5 2 3 7 1 7 5 1 1 3

The combinatorics Young tableaux of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A Young tableau is a filling of a Young diagram with The poset L(m, n) positive integers. Young tableaux Young tableaux Counting tableaux

“The tableau is on the diagram λ” or “λ is the shape of the SST as a monoid tableau.” Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Young tableaux of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A Young tableau is a filling of a Young diagram with The poset L(m, n) positive integers. Young tableaux Young tableaux Counting tableaux

“The tableau is on the diagram λ” or “λ is the shape of the SST as a monoid tableau.” Skew tableaux Jeu de taquin The product on SST Examples: Factoring Young tableaux Littlewood-Richardson numbers 1 10 2 1 4 The Pieri rule 5 2 3 7 The Littlewood-Richardson 1 7 5 rule 1 1 3 Examples:

1 2 3 7 9 1 2 4 5 4 6 10 3 5 8

The combinatorics Standard tableaux of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A Young tableau on the shape λ is called standard if The poset L(m, n) ▶ the filling consists of 1,..., |λ|, each appearing exactly Young tableaux Young tableaux once, Counting tableaux ▶ SST as a monoid the filling is weakly increasing across each rowand Skew tableaux Jeu de taquin ▶ strictly increasing down each column. The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Standard tableaux of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A Young tableau on the shape λ is called standard if The poset L(m, n) ▶ the filling consists of 1,..., |λ|, each appearing exactly Young tableaux Young tableaux once, Counting tableaux ▶ SST as a monoid the filling is weakly increasing across each rowand Skew tableaux Jeu de taquin ▶ strictly increasing down each column. The product on SST Factoring Young tableaux Littlewood-Richardson Examples: numbers The Pieri rule The Littlewood-Richardson 1 2 3 7 9 rule 1 2 4 5 4 6 10 3 5 8 The combinatorics Standard tableaux of Young tableaux John A. Miller

More Examples: Young Diagrams Partitions Young diagrams The poset L(m, n) 1 Young tableaux Young tableaux 1 6 10 13 15 2 Counting tableaux 1 2 6 SST as a monoid 2 7 11 14 3 Skew tableaux 3 5 10 Jeu de taquin 3 8 12 4 The product on SST 4 8 Factoring Young 4 9 5 tableaux 7 9 Littlewood-Richardson 5 6 numbers The Pieri rule The Littlewood-Richardson 7 rule

1 2 3 4 5 6 7 Examples:

1 2 2 3 4 2 2 4 7 2 4 5 3 4 5

The combinatorics Semi-standard tableaux of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A Young tableau is called semi-standard if the filling is The poset L(m, n) ▶ weakly increasing across each row and Young tableaux Young tableaux ▶ strictly increasing down each column Counting tableaux SST as a monoid Skew tableaux SST = the set of all semi-standard Young tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Semi-standard tableaux of Young tableaux John A. Miller

Factoring Young Examples: tableaux Littlewood-Richardson numbers The Pieri rule 1 2 2 3 4 The Littlewood-Richardson 2 2 4 7 rule 2 4 5 3 4 5 The combinatorics Semi-standard tableaux of Young tableaux John A. Miller

More Examples: Young Diagrams Partitions Young diagrams The poset L(m, n) 1 Young tableaux Young tableaux 1 1 1 2 2 2 Counting tableaux 1 1 1 SST as a monoid 2 2 3 3 3 Skew tableaux 2 2 2 Jeu de taquin 3 3 6 5 The product on SST 3 3 Factoring Young 4 7 6 tableaux 4 4 Littlewood-Richardson 5 7 numbers The Pieri rule The Littlewood-Richardson 8 rule

1 1 1 2 2 2 4 It is clear fλ and dλ(m) are finite, but it is not so obvious how to find formulas.

Turns out to be related to hook length.

The combinatorics Counting standard and semi-standard tableaux of Young tableaux John A. Miller

For a given shape λ, let Young Diagrams Partitions Young diagrams fλ = # {standard tableaux of shape λ} The poset L(m, n) Young tableaux Young tableaux and Counting tableaux

SST as a monoid d (m) = # {SST of shape λ with entries in 1,..., m} . Skew tableaux λ Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Counting standard and semi-standard tableaux of Young tableaux John A. Miller

For a given shape λ, let Young Diagrams Partitions Young diagrams fλ = # {standard tableaux of shape λ} The poset L(m, n) Young tableaux Young tableaux and Counting tableaux

SST as a monoid d (m) = # {SST of shape λ with entries in 1,..., m} . Skew tableaux λ Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers It is clear fλ and dλ(m) are finite, but it is not so obvious The Pieri rule The Littlewood-Richardson how to find formulas. rule

Turns out to be related to hook length. Each (i, j) in λ determines a hook, which is the set of boxes containing (i, j), all boxes in row i to the right of (i, j), and all boxes in column j below (i, j).

The combinatorics Hook length of Young tableaux John A. Miller

For given a Young diagram λ, let (i, j) denote the box in row Young Diagrams Partitions i and column j. Young diagrams The poset L(m, n)

Young tableaux Young tableaux 1 Counting tableaux SST as a monoid 2 Skew tableaux Jeu de taquin 3 The product on SST Factoring Young 4 tableaux Littlewood-Richardson 5 numbers The Pieri rule The Littlewood-Richardson 1 2 3 4 5 6 rule The combinatorics Hook length of Young tableaux John A. Miller

For given a Young diagram λ, let (i, j) denote the box in row Young Diagrams Partitions i and column j. Young diagrams The poset L(m, n)

Each (i, j) in λ determines a hook, which is the set of boxes containing (i, j), all boxes in row i to the right of (i, j), and all boxes in column j below (i, j). The combinatorics Hook length of Young tableaux John A. Miller

For given a Young diagram λ, let (i, j) denote the box in row Young Diagrams Partitions i and column j. Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule e.g. (2, 2) The combinatorics Hook length of Young tableaux John A. Miller

For given a Young diagram λ, let (i, j) denote the box in row Young Diagrams Partitions i and column j. Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule e.g. (2, 2) The combinatorics Hook length of Young tableaux John A. Miller

For given a Young diagram λ, let (i, j) denote the box in row Young Diagrams Partitions i and column j. Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule e.g. (2, 2) and its hook h(2, 2) = 5

The combinatorics Hook length of Young tableaux John A. Miller

The hook length of a box (i, j), denoted h(i, j), is the number Young Diagrams Partitions of boxes in its hook. Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Hook length of Young tableaux John A. Miller

The hook length of a box (i, j), denoted h(i, j), is the number Young Diagrams Partitions of boxes in its hook. Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule h(2, 2) = 5 The combinatorics Hook length of Young tableaux John A. Miller

The hook length of a box (i, j), denoted h(i, j), is the number Young Diagrams Partitions of boxes in its hook. Young diagrams The poset L(m, n)

Young tableaux Young tableaux 10 8 6 5 2 1 Counting tableaux SST as a monoid 7 5 3 2 Skew tableaux Jeu de taquin 6 4 2 1 The product on SST Factoring Young 3 1 tableaux Littlewood-Richardson 1 numbers The Pieri rule The Littlewood-Richardson rule Labeling each box (i, j) with its hook length h(i, j). The combinatorics The number of standard tableaux on λ of Young tableaux John A. Miller

Young Diagrams Theorem (Frame, Robinson, Thrall (1954)) Partitions Young diagrams If λ is a Young diagram with n boxes, then the number of The poset L(m, n) standard tableaux on the shape λ is Young tableaux Young tableaux Counting tableaux ∏ n! SST as a monoid fλ = . Skew tableaux h(i, j) Jeu de taquin (i,j)∈λ The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics The number of standard tableaux on λ of Young tableaux John A. Miller

Young Diagrams Theorem (Frame, Robinson, Thrall (1954)) Partitions Young diagrams If λ is a Young diagram with n boxes, then the number of The poset L(m, n) standard tableaux on the shape λ is Young tableaux Young tableaux Counting tableaux ∏ n! SST as a monoid fλ = . Skew tableaux h(i, j) Jeu de taquin (i,j)∈λ The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule e.g. λ = (3, 2, 1), |λ| = 6 The Littlewood-Richardson rule The combinatorics The number of standard tableaux on λ of Young tableaux John A. Miller

Young Diagrams Theorem (Frame, Robinson, Thrall (1954)) Partitions Young diagrams If λ is a Young diagram with n boxes, then the number of The poset L(m, n) standard tableaux on the shape λ is Young tableaux Young tableaux Counting tableaux ∏ n! SST as a monoid fλ = . Skew tableaux h(i, j) Jeu de taquin (i,j)∈λ The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule e.g. λ = (3, 2, 1), |λ| = 6 The Littlewood-Richardson rule

5 3 1 3 1 1 The combinatorics The number of standard tableaux on λ of Young tableaux John A. Miller

Young Diagrams Theorem (Frame, Robinson, Thrall (1954)) Partitions Young diagrams If λ is a Young diagram with n boxes, then the number of The poset L(m, n) standard tableaux on the shape λ is Young tableaux Young tableaux Counting tableaux ∏ n! SST as a monoid fλ = . Skew tableaux h(i, j) Jeu de taquin (i,j)∈λ The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule e.g. λ = (3, 2, 1), |λ| = 6 The Littlewood-Richardson rule 6! 5 3 1 f = = 24 λ 5 · 32 · 13 3 1 1 The combinatorics The number of standard tableaux on λ of Young tableaux John A. Miller

σ = (6, 4, 4, 2, 1), |λ| = 17 Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics The number of standard tableaux on λ of Young tableaux John A. Miller

σ = (6, 4, 4, 2, 1), |λ| = 17 Young Diagrams Partitions Young diagrams The poset L(m, n) 10 8 6 5 2 1 Young tableaux Young tableaux 7 5 3 2 Counting tableaux SST as a monoid 6 4 2 1 Skew tableaux Jeu de taquin 3 1 The product on SST Factoring Young 1 tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics The number of standard tableaux on λ of Young tableaux John A. Miller

σ = (6, 4, 4, 2, 1), |λ| = 17 Young Diagrams Partitions Young diagrams The poset L(m, n) 10 8 6 5 2 1 Young tableaux Young tableaux 7 5 3 2 Counting tableaux SST as a monoid 6 4 2 1 Skew tableaux Jeu de taquin 3 1 The product on SST Factoring Young 1 tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule 17! f = = 2, 450, 448 σ 10 · 8 · 7 · 62 · 52 · 4 · 32 · 23 · 14 The numerators m + j − i are obtained by putting the numbers m on the diagonal of λ and putting the numbers m p in the boxes that are p steps above or below the diagonal.

The combinatorics The number of semi-standard tableaux on λ of Young tableaux John A. Miller

Young Diagrams Theorem (Stanley (1971)) Partitions Young diagrams If λ is a Young diagram, then the number of semi-standard The poset L(m, n) tableaux with entries in 1,..., m on the shape λ is Young tableaux Young tableaux Counting tableaux ∏ m + j − i d (m) = . SST as a monoid λ h(i, j) Skew tableaux (i,j)∈λ Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics The number of semi-standard tableaux on λ of Young tableaux John A. Miller

Factoring Young tableaux Littlewood-Richardson numbers The numerators m + j − i are obtained by putting the The Pieri rule The Littlewood-Richardson numbers m on the diagonal of λ and putting the numbers rule m p in the boxes that are p steps above or below the diagonal. The combinatorics The number of semi-standard tableaux on λ of Young tableaux John A. Miller

e.g. λ = (4, 2, 2, 1), m = 5 Young Diagrams Partitions Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics The number of semi-standard tableaux on λ of Young tableaux John A. Miller

e.g. λ = (4, 2, 2, 1), m = 5 Young Diagrams Partitions Young diagrams Label λ with the terms in the The poset L(m, n) numerator, m + j − i Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics The number of semi-standard tableaux on λ of Young tableaux John A. Miller

e.g. λ = (4, 2, 2, 1), m = 5 Young Diagrams Partitions Young diagrams Label λ with the terms in the The poset L(m, n) numerator, m + j − i Young tableaux Young tableaux Counting tableaux 5 6 7 8 SST as a monoid Skew tableaux 4 5 Jeu de taquin The product on SST 3 4 Factoring Young tableaux Littlewood-Richardson 2 numbers The Pieri rule The Littlewood-Richardson rule The combinatorics The number of semi-standard tableaux on λ of Young tableaux John A. Miller

e.g. λ = (4, 2, 2, 1), m = 5 Young Diagrams Partitions Young diagrams Label λ with the terms in the Label λ with the terms in the The poset L(m, n) numerator, m + j − i denominator, h(i, j) Young tableaux Young tableaux Counting tableaux 5 6 7 8 7 5 2 1 SST as a monoid Skew tableaux 4 5 Jeu de taquin 4 2 The product on SST 3 4 3 1 Factoring Young tableaux Littlewood-Richardson 2 1 numbers The Pieri rule The Littlewood-Richardson rule The combinatorics The number of semi-standard tableaux on λ of Young tableaux John A. Miller

e.g. λ = (4, 2, 2, 1), m = 5 Young Diagrams Partitions Young diagrams Label λ with the terms in the Label λ with the terms in the The poset L(m, n) numerator, m + j − i denominator, h(i, j) Young tableaux Young tableaux Counting tableaux 5 6 7 8 7 5 2 1 SST as a monoid Skew tableaux 4 5 Jeu de taquin 4 2 The product on SST 3 4 3 1 Factoring Young tableaux Littlewood-Richardson 2 1 numbers The Pieri rule The Littlewood-Richardson rule 8 · 7 · 6 · 52 · 42 · 3 · 2 d (5) = = 480 λ 7 · 5 · 4 · 3 · 22 · 13 Rest of talk

▶ View SST as a monoid ▶ ν Littlewood-Richardson numbers cλµ

The combinatorics Why Young tableaux? of Young tableaux John A. Miller

▶ Young Diagrams Inherently interesting combinatorial objects, relationship Partitions with integer partitions Young diagrams The poset L(m, n) ▶ Relationship with symmetric functions Young tableaux ▶ Young tableaux Combinatorics of Young tableaux ⇝ representation Counting tableaux theory SST as a monoid Skew tableaux ▶ Parametrize all irreducible representations of Sn (which Jeu de taquin The product on SST have dimension Fλ) ▶ Factoring Young Parametrize all polynomial irreducible reps of GL(n) tableaux ▶ Littlewood-Richardson Schubert calculus on Grassmannians and flag varieties numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Why Young tableaux? of Young tableaux John A. Miller

▶ View SST as a monoid ▶ ν Littlewood-Richardson numbers cλµ So to view SST as a monoid, we need to define a product

· : SST × SST → SST

such that ▶ T · (U · V) = (T · U) · V for all T, U, V ∈ SST ▶ ∃ a unit e ∈ SST, i.e. e · T = T · e = T for all T ∈ SST

To define this multiplication, we need to first introduce skew tableaux and jeu de taquin (“teasing game”).

The combinatorics Monoids of Young tableaux John A. Miller

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule To define this multiplication, we need to first introduce skew tableaux and jeu de taquin (“teasing game”).

The combinatorics Monoids of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A monoid is a set with an associative multiplication and a The poset L(m, n) unit. Young tableaux Young tableaux Think: like a group, but without inverses. Counting tableaux SST as a monoid Skew tableaux So to view SST as a monoid, we need to define a product Jeu de taquin The product on SST · × → Factoring Young : SST SST SST tableaux Littlewood-Richardson numbers such that The Pieri rule The Littlewood-Richardson ▶ T · (U · V) = (T · U) · V for all T, U, V ∈ SST rule ▶ ∃ a unit e ∈ SST, i.e. e · T = T · e = T for all T ∈ SST The combinatorics Monoids of Young tableaux John A. Miller

To define this multiplication, we need to first introduce skew tableaux and jeu de taquin (“teasing game”). Example: If µ = (3, 2) and λ = (5, 3, 3, 1), then

λ/µ = =

The combinatorics Skew diagrams of Young tableaux John A. Miller

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Skew diagrams of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams For Young diagrams λ and µ with µ ⊂ λ, the skew diagram The poset L(m, n) λ/µ is the diagram obtained by removing µ from λ. Young tableaux Young tableaux Counting tableaux Example: SST as a monoid Skew tableaux Jeu de taquin If µ = (3, 2) and λ = (5, 3, 3, 1), then The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule

λ/µ = = Examples:

λ/λ = ·

2 3 1 1 1 3 1 1 2 3 1 1 9 1

The combinatorics Skew tableaux of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A skew tableau is a skew diagram filled with positive The poset L(m, n) integers. Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Skew tableaux of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A skew tableau is a skew diagram filled with positive The poset L(m, n) integers. Young tableaux Young tableaux Counting tableaux Examples: SST as a monoid Skew tableaux Jeu de taquin λ/λ = · The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson 2 3 1 1 rule 1 3 1 1 2 3 1 1 9 1 Examples:

5 7 3 4 1 2 1 2 3 6 3 4 1 2

The combinatorics Standard skew tableaux of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A skew tableau with n boxes is called standard if The poset L(m, n) 1. the filling consists of 1,..., n, each appearing exactly Young tableaux Young tableaux once, Counting tableaux SST as a monoid 2. the filling is weakly increasing across each rowand Skew tableaux Jeu de taquin 3. strictly increasing down each column The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Standard skew tableaux of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A skew tableau with n boxes is called standard if The poset L(m, n) 1. the filling consists of 1,..., n, each appearing exactly Young tableaux Young tableaux once, Counting tableaux SST as a monoid 2. the filling is weakly increasing across each rowand Skew tableaux Jeu de taquin 3. strictly increasing down each column The product on SST Factoring Young tableaux Littlewood-Richardson Examples: numbers The Pieri rule The Littlewood-Richardson rule 5 7 3 4 1 2 1 2 3 6 3 4 1 2 Examples:

2 3 1 1 1 1 2 2 1 2 2 3 5 1 3 1 2

The combinatorics Semi-standard skew tableaux of Young tableaux John A. Miller

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Semi-standard skew tableaux of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams A skew tableau is called semi-standard if the filling is The poset L(m, n) 1. weakly increasing across each row and Young tableaux Young tableaux 2. strictly increasing down each column Counting tableaux SST as a monoid Skew tableaux Jeu de taquin Examples: The product on SST Factoring Young tableaux Littlewood-Richardson numbers 2 3 1 1 The Pieri rule The Littlewood-Richardson 1 1 2 2 rule 1 2 2 3 5 1 3 1 2 The combinatorics Skew tableaux of Young tableaux John A. Miller

For a skew tableaux λ/µ, Young Diagrams Partitions ▶ an inside corner is a box in µ such that the boxes Young diagrams The poset L(m, n)

below and to the right are not in µ Young tableaux ▶ Young tableaux an outside corner is a box in λ such that the boxes Counting tableaux below and to the right are not in λ SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Skew tableaux of Young tableaux John A. Miller

For a skew tableaux λ/µ, Young Diagrams Partitions ▶ an inside corner is a box in µ such that the boxes Young diagrams The poset L(m, n)

Factoring Young 2 3 tableaux Littlewood-Richardson numbers 1 The Pieri rule The Littlewood-Richardson 1 2 4 rule 3 The combinatorics Skew tableaux of Young tableaux John A. Miller

For a skew tableaux λ/µ, Young Diagrams Partitions ▶ an inside corner is a box in µ such that the boxes Young diagrams The poset L(m, n)

Factoring Young 2 3 tableaux Littlewood-Richardson numbers 1 The Pieri rule The Littlewood-Richardson 1 2 4 rule 3 The combinatorics Skew tableaux of Young tableaux John A. Miller

For a skew tableaux λ/µ, Young Diagrams Partitions ▶ an inside corner is a box in µ such that the boxes Young diagrams The poset L(m, n)

Factoring Young 2 3 tableaux Littlewood-Richardson numbers 1 The Pieri rule The Littlewood-Richardson 1 2 4 rule 3 The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Given a skew tableaux S, we can form a tableaux called the Young Diagrams Partitions rectification of S, Rect(S), using a method called sliding. Young diagrams The poset L(m, n)

Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule 1. Pick an inside corner of S, 2. Consider the boxes to the right and below the empty box 3. Slide the box with the lesser entry into the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner

The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 2 3 Partitions Young diagrams 1 The poset L(m, n) Young tableaux 1 2 4 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule 2. Consider the boxes to the right and below the empty box 3. Slide the box with the lesser entry into the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner

The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 2 3 Partitions Young diagrams 1 The poset L(m, n) Young tableaux 1 2 4 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule 2. Consider the boxes to the right and below the empty box 3. Slide the box with the lesser entry into the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner

The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 2 3 Partitions Young diagrams 1 The poset L(m, n) Young tableaux 1 2 4 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule The Littlewood-Richardson 3. Slide the box with the lesser entry into the empty box rule 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner

The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 2 3 Partitions Young diagrams 1 The poset L(m, n) Young tableaux 1 2 4 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule The Littlewood-Richardson 3. Slide the box with the lesser entry into the empty box rule 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner

The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 2 3 Partitions Young diagrams 1 The poset L(m, n) Young tableaux 1 2 4 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule The Littlewood-Richardson 3. Slide the box with the lesser entry into the empty box rule The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 2 3 Partitions Young diagrams 1 The poset L(m, n) Young tableaux 1 2 4 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule The Littlewood-Richardson 3. Slide the box with the lesser entry into the empty box rule 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 2 3 Partitions Young diagrams 1 The poset L(m, n) Young tableaux 1 2 4 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule The Littlewood-Richardson 3. Slide the box with the lesser entry into the empty box rule ▶ If there is only one option, slide it into the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 2 3 Partitions Young diagrams 1 4 The poset L(m, n) Young tableaux 1 2 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule The Littlewood-Richardson 3. Slide the box with the lesser entry into the empty box rule ▶ If there is only one option, slide it into the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 2 3 Partitions Young diagrams 1 4 The poset L(m, n) Young tableaux 1 2 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule The Littlewood-Richardson 3. Slide the box with the lesser entry into the empty box rule ▶ If there is only one option, slide it into the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 2 3 Partitions Young diagrams 1 4 The poset L(m, n) Young tableaux 1 2 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule 3. Slide the box with the lesser entry into the empty box The Littlewood-Richardson rule ▶ If there is only one option, slide it into the empty box ▶ If both options have the same value, choose the box below the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 2 3 Partitions Young diagrams 1 1 4 The poset L(m, n) Young tableaux 2 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule 3. Slide the box with the lesser entry into the empty box The Littlewood-Richardson rule ▶ If there is only one option, slide it into the empty box ▶ If both options have the same value, choose the box below the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 1 2 3 Partitions Young diagrams 1 4 The poset L(m, n) Young tableaux 2 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule 3. Slide the box with the lesser entry into the empty box The Littlewood-Richardson rule ▶ If there is only one option, slide it into the empty box ▶ If both options have the same value, choose the box below the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 1 1 2 3 Partitions Young diagrams 4 The poset L(m, n) Young tableaux 2 Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule 3. Slide the box with the lesser entry into the empty box The Littlewood-Richardson rule ▶ If there is only one option, slide it into the empty box ▶ If both options have the same value, choose the box below the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 1 1 2 3 Partitions Young diagrams 2 4 The poset L(m, n) Young tableaux Young tableaux Counting tableaux 3 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule 3. Slide the box with the lesser entry into the empty box The Littlewood-Richardson rule ▶ If there is only one option, slide it into the empty box ▶ If both options have the same value, choose the box below the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 1 1 2 3 Partitions Young diagrams 2 4 The poset L(m, n) Young tableaux 3 Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule 3. Slide the box with the lesser entry into the empty box The Littlewood-Richardson rule ▶ If there is only one option, slide it into the empty box ▶ If both options have the same value, choose the box below the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Rect(S) via jeu de taquin (teasing game) of Young tableaux John A. Miller

Young Diagrams 1 1 2 3 Partitions Young diagrams 2 4 The poset L(m, n) Young tableaux 3 Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin The product on SST

Facts Factoring Young tableaux ▶ Littlewood-Richardson Rect(S) is a tableaux. numbers The Pieri rule ▶ The Littlewood-Richardson If S is semi-standard, so is Rect(S) rule ▶ Rect(S) does not depend on the choice of inside corners.

We will use the jeu de taquin to define the product · on SST! The combinatorics Product on SST of Young tableaux John A. Miller

Given T, U ∈ SST, form the skew tableau T ⋆ U by adjoining Young Diagrams Partitions T and U in the following way: Young diagrams The poset L(m, n) 1 1 2 3 Young tableaux 1 2 2 Young tableaux T = 2 3 4 U = Counting tableaux 3 SST as a monoid 3 4 Skew tableaux Jeu de taquin The product on SST

Factoring Young tableaux 1 2 2 Littlewood-Richardson numbers 3 The Pieri rule The Littlewood-Richardson rule T ⋆ U = 1 1 2 3 2 3 4 3 4 The combinatorics Product on SST of Young tableaux John A. Miller

Young Diagrams Definition Partitions Young diagrams The poset L(m, n) T · U := Rect(T ⋆ U) Young tableaux Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin Let’s find T · U for the previous example. The product on SST Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Product on SST using jeu de taquin of Young tableaux John A. Miller

Young Diagrams 1 2 2 Partitions Young diagrams 3 The poset L(m, n) Young tableaux T ⋆ U = 1 1 2 3 Young tableaux Counting tableaux 2 3 4 SST as a monoid Skew tableaux 3 4 Jeu de taquin The product on SST

Factoring Young tableaux Littlewood-Richardson numbers The Pieri rule The Littlewood-Richardson rule The combinatorics Product on SST using jeu de taquin of Young tableaux John A. Miller

Young Diagrams 1 2 2 Partitions Young diagrams 3 The poset L(m, n) Young tableaux 1 1 2 3 Young tableaux Counting tableaux 2 3 4 SST as a monoid Skew tableaux 3 4 Jeu de taquin The product on SST

Factoring Young 1. Pick an inside corner of S, an empty box tableaux Littlewood-Richardson 2. Consider the boxes to the right and below the empty numbers box The Pieri rule The Littlewood-Richardson 3. Slide the box with the lesser entry into the empty box rule ▶ If there is only one option, slide it into the empty box ▶ If both options have the same value, choose the box below the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Product on SST using jeu de taquin of Young tableaux John A. Miller

Young Diagrams 1 2 2 Partitions Young diagrams 3 The poset L(m, n) Young tableaux 1 1 2 3 Young tableaux Counting tableaux 2 3 4 SST as a monoid Skew tableaux 3 4 Jeu de taquin The product on SST

Young Diagrams 1 2 2 Partitions Young diagrams 3 3 The poset L(m, n) Young tableaux 1 1 2 Young tableaux Counting tableaux 2 3 4 SST as a monoid Skew tableaux 3 4 Jeu de taquin The product on SST

Young Diagrams 1 2 2 Partitions Young diagrams 2 3 3 The poset L(m, n) Young tableaux 1 1 Young tableaux Counting tableaux 2 3 4 SST as a monoid Skew tableaux 3 4 Jeu de taquin The product on SST

Young Diagrams 1 2 2 Partitions Young diagrams 2 3 3 The poset L(m, n) Young tableaux 1 1 4 Young tableaux Counting tableaux 2 3 SST as a monoid Skew tableaux 3 4 Jeu de taquin The product on SST

Young Diagrams 1 2 2 Partitions Young diagrams 1 2 3 3 The poset L(m, n) Young tableaux 1 4 Young tableaux Counting tableaux 2 3 SST as a monoid Skew tableaux 3 4 Jeu de taquin The product on SST

Young Diagrams 1 2 2 Partitions Young diagrams 1 2 3 3 The poset L(m, n) Young tableaux 1 3 4 Young tableaux Counting tableaux 2 SST as a monoid Skew tableaux 3 4 Jeu de taquin The product on SST

Young Diagrams 1 2 2 Partitions Young diagrams 1 2 3 3 The poset L(m, n) Young tableaux 1 3 4 Young tableaux Counting tableaux 2 4 SST as a monoid Skew tableaux 3 Jeu de taquin The product on SST

Young Diagrams 1 1 2 2 Partitions Young diagrams 2 3 3 The poset L(m, n) Young tableaux 1 3 4 Young tableaux Counting tableaux 2 4 SST as a monoid Skew tableaux 3 Jeu de taquin The product on SST

Young Diagrams 1 1 2 2 Partitions Young diagrams 1 2 3 3 The poset L(m, n) Young tableaux 3 4 Young tableaux Counting tableaux 2 4 SST as a monoid Skew tableaux 3 Jeu de taquin The product on SST

Young Diagrams 1 1 2 2 Partitions Young diagrams 1 2 3 3 The poset L(m, n) Young tableaux 2 3 4 Young tableaux Counting tableaux 4 SST as a monoid Skew tableaux 3 Jeu de taquin The product on SST

Young Diagrams 1 1 2 2 Partitions Young diagrams 1 2 3 3 The poset L(m, n) Young tableaux 2 3 4 Young tableaux Counting tableaux 3 4 SST as a monoid Skew tableaux Jeu de taquin The product on SST

Young Diagrams 1 1 2 2 Partitions Young diagrams 1 2 3 3 The poset L(m, n) Young tableaux 2 3 4 Young tableaux Counting tableaux 3 4 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule 3. Slide the box with the lesser entry into the empty box The Littlewood-Richardson rule ▶ If there is only one option, slide it into the empty box ▶ If both options have the same value, choose the box below the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Product on SST using jeu de taquin of Young tableaux John A. Miller

Young Diagrams 1 1 1 2 2 Partitions Young diagrams 2 3 3 The poset L(m, n) Young tableaux 2 3 4 Young tableaux Counting tableaux 3 4 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule 3. Slide the box with the lesser entry into the empty box The Littlewood-Richardson rule ▶ If there is only one option, slide it into the empty box ▶ If both options have the same value, choose the box below the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Product on SST using jeu de taquin of Young tableaux John A. Miller

Young Diagrams 1 1 1 2 2 Partitions Young diagrams 2 2 3 3 The poset L(m, n) Young tableaux 3 4 Young tableaux Counting tableaux 3 4 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule 3. Slide the box with the lesser entry into the empty box The Littlewood-Richardson rule ▶ If there is only one option, slide it into the empty box ▶ If both options have the same value, choose the box below the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Product on SST using jeu de taquin of Young tableaux John A. Miller

Young Diagrams 1 1 1 2 2 Partitions Young diagrams 2 2 3 3 The poset L(m, n) Young tableaux 3 3 4 Young tableaux Counting tableaux 4 SST as a monoid Skew tableaux Jeu de taquin 1. Pick an inside corner of S, an empty box The product on SST Factoring Young 2. Consider the boxes to the right and below the empty tableaux Littlewood-Richardson box numbers The Pieri rule 3. Slide the box with the lesser entry into the empty box The Littlewood-Richardson rule ▶ If there is only one option, slide it into the empty box ▶ If both options have the same value, choose the box below the empty box 4. Repeat steps 2 & 3 until you have moved the empty box to an outside corner, then delete the box and repeat steps 1 - 4 The combinatorics Product on SST using jeu de taquin of Young tableaux John A. Miller

Young Diagrams 1 1 2 3 Partitions 1 2 2 Young diagrams T = 2 3 4 U = The poset L(m, n) 3 Young tableaux 3 4 Young tableaux Counting tableaux

SST as a monoid Skew tableaux Jeu de taquin T ⋆ U = The product on SST T · U := Rect(T ⋆ U) = Factoring Young 1 2 2 tableaux Littlewood-Richardson 3 numbers 1 1 1 2 2 The Pieri rule The Littlewood-Richardson 1 1 2 3 2 2 3 3 rule 2 3 4 3 3 4 3 4 4 The combinatorics SST as a monoid of Young tableaux John A. Miller

Remember the goal: view SST as a monoid. Young Diagrams Partitions Young diagrams The empty diagram in SST serves as the unit: The poset L(m, n) Young tableaux Young tableaux 1 1 2 3 Counting tableaux ∅ · ⇒ ∅ SST as a monoid T = 2 3 4 , = = T ⋆ = T Skew tableaux Jeu de taquin 3 4 The product on SST Factoring Young tableaux Littlewood-Richardson numbers T · ∅ = Rect(T ⋆ ∅) = Rect(T) = T The Pieri rule The Littlewood-Richardson rule Similarly,

∅ · T = Rect(∅ ⋆ T) = Rect(T) = T The combinatorics SST as a monoid of Young tableaux John A. Miller

Claim Young Diagrams Partitions · × → Young diagrams : YT YT YT is associative. The poset L(m, n)

Young tableaux Idea: Choice of inside corners doesn’t matter. Young tableaux Counting tableaux

SST as a monoid 1 2 1 Skew tableaux T = U = 1 2 V = Jeu de taquin 2 2 The product on SST Factoring Young tableaux 1 Littlewood-Richardson numbers The Pieri rule 2 The Littlewood-Richardson rule T ⋆ U ⋆ V = 1 2 1 2 2

(T · U) · V = T · (U · V) The combinatorics SST as a monoid of Young tableaux John A. Miller