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 I Partitions syzygies of modules of covariants Young diagrams The poset L(m; n) I minimal free resolutions in classical invariant theory Young tableaux I 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 I A flavor of the combinatorics of Young tableaux I 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 I tableaux Born Widnes, Lancashire, England 1873 Littlewood-Richardson numbers I Introduced Young tableaux in 1900 while working in The Pieri rule The Littlewood-Richardson classical invariant theory rule I Work extended by Frobenius, Schur, Weyl in early 1900s I Only finitely many λi =6 0 I Each λi =6 0 is a part of λ I λ ` n or jλj = n λ = (4; 2; 1; 0; 0;:::) = (4; 2; 1; 0; 0) = (4; 2; 1) jλj = 7 The combinatorics Integer Partitions of Young tableaux John A. Miller Young Diagrams Definition Partitions Young diagrams A partition of n 2 N is a sequence of integers The poset L(m; n) Young tableaux Young tableaux λ = (λ1; λ2;:::) Counting tableaux X 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) jλj = 7 The combinatorics Integer Partitions of Young tableaux John A. Miller Young Diagrams Definition Partitions Young diagrams A partition of n 2 N is a sequence of integers The poset L(m; n) Young tableaux Young tableaux λ = (λ1; λ2;:::) Counting tableaux X SST as a monoid where λ1 ≥ λ2 ≥ · · · ≥ 0 and λi = n. Skew tableaux Jeu de taquin The product on SST Factoring Young tableaux I Littlewood-Richardson Only finitely many λi =6 0 numbers The Pieri rule I 6 The Littlewood-Richardson Each λi = 0 is a part of λ rule I λ ` n or jλj = n The combinatorics Integer Partitions of Young tableaux John A. Miller Young Diagrams Definition Partitions Young diagrams A partition of n 2 N is a sequence of integers The poset L(m; n) Young tableaux Young tableaux λ = (λ1; λ2;:::) Counting tableaux X SST as a monoid where λ1 ≥ λ2 ≥ · · · ≥ 0 and λi = n. Skew tableaux Jeu de taquin The product on SST Factoring Young tableaux I Littlewood-Richardson Only finitely many λi =6 0 numbers The Pieri rule I 6 The Littlewood-Richardson Each λi = 0 is a part of λ rule I λ ` n or jλj = n λ = (4; 2; 1; 0; 0;:::) = (4; 2; 1; 0; 0) = (4; 2; 1) jλj = 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.