Semistandard Young Tableaux Polytopes. Sara Solhjem Joint work with Jessica Striker. April 9, 2017

Transcription

1 Semistandard Young Tableaux Polytopes Sara Solhjem Joint work with Jessica Striker North Dakota State University Graduate Student Combinatorics Conference 217 April 9, 217 Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

2 Main Topics 1 Background 2 Polytope with shape λ and max entry n 3 Polytope of m n sign matrices 4 Comparing P(λ, n) and P(m, n) 5 Future Connections Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

3 Main Topics 1 Background 2 Polytope with shape λ and max entry n 3 Polytope of m n sign matrices 4 Comparing P(λ, n) and P(m, n) 5 Future Connections Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

4 Semistandard Young Tableaux Definition A Young diagram λ is a collection of boxes, or cells, arranged in left-justified rows, with a weakly decreasing number of boxes in each row. Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

5 Semistandard Young Tableaux Definition A semistandard Young tableau, denoted SSYT, is defined as a filling of a Young diagram such that the rows are weakly increasing and the columns are strictly increasing Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

6 SSYT(m,n) Definition Let SSYT (m, n) denote the set of semistandard Young tableaux with at most m columns and entries at most n SSYT (6, 7) Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

7 Notation with the SSYT(m,n) λ is the shape using row length. [6, 3, 3, 1] λ i is the length of a row in the tableau. λ 1 = 6, λ 2 = 3, λ 3 = 3, λ 4 = 1 l i is the length of a column read right to left. l =, l 1 = 1, l 2 = 1, l 3 = 1, l 4 = 3, l 5 = 3, l 6 = 4 l(λ) is the number of rows in the tableau. 4 Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

8 m n Sign Matrices Definition (Aval 28) Sign matrices are m n matrices with the following properties: entries { 1,, 1} the partial sums of the columns are or 1 the partial sums of the rows are always nonnegative Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

9 Bijection with semistandard Young Tableau Aval also showed that sign matrices are in bijection with semistandard Young Tableau Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

10 SSYT(λ, n) Definition Let SSYT (λ, n) denote the set of semistandard Young tableaux of shape λ and entries at most n. Proposition (Stanley 1971) The number of SSYT of shape λ with maximum entry of n, is given by the hook-content formula: u λ n + c(u) h(u) Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

11 Sign Matrices M(λ, n) Definition Let M(λ, n) be the set of λ 1 n sign matrices M = (M ij ) such that: n M ij = l i l i 1 for all 1 i λ 1, j=1 Call M(λ, n) the set of sign matrices of shape λ and content at most n. Proposition (S. and Striker) M(λ, n) is in explicit bijection with SSYT (λ, n). Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

12 Tableau shape λ and max entry at most n Notice the correspondence between the shape of the tableau and the size of the matrix: the number of rows of the matrix and the length of the first row of the tableau, λ 1. the number of columns in the matrix and the max entry in the tableau. the total sum of each row and l i l i Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

13 What is a Polytope? A polytope can be defined in two equivalent ways: As the convex hull of a finite set of points {x 1, x 2,..., x n }, that is, the set of all expressions of n n the form µ i x i where µ i = 1 and all the µ i are i=1 nonnegative. i=1 As the bounded intersection of finitely many closed halfspaces. Thus a polytope can be specified by a set of points or by a set of linear inequalities. Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

14 Main Topics 1 Background 2 Polytope with shape λ and max entry n 3 Polytope of m n sign matrices 4 Comparing P(λ, n) and P(m, n) 5 Future Connections Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

15 The P(λ, n) Polytope Definition Let P(λ, n) be the polytope defined as the convex hull of all the matrices in M(λ, n). P([2, 2], 3) = { ( ) ( ) µ 1 + µ 2 ( ) ( ) µ 4 + µ ( ) µ µ 6 ( )} 6 where µ i = 1. i=1 Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

16 The P(λ, n) Polytope Theorem (S. and Striker) The dimension of the P(λ, n) polytope is λ 1 (n 1) for 1 l(λ) < n. When l(λ) = n the dimension is (λ 1 λ n )(n 1) Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

17 Vertex Theorem Theorem (S. and Striker) Each of the SSYT forms a vertex in the P(λ, n) polytope. c 1 c 2... c n c 11 c 12 c 3 r 1 r 11 r 12 r 1n X 11 X 12 X 1n c 1n r 2 r21 r 22 r 2n X 21 X 22 X 2n.... rm r m1 r m2 r mn X m1 X m2 X mn c m1 c m2 c m3 The graph of partial sums is in bijection with P(λ, n).... c mn Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

18 Partial sum graphs of all six vertices in P([2, 2], 3) (a) (b) (c) (d) 1 1 (e) 1 (f) 1 Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

19 Sketch of the vertex proof [ ] The hyperplane that separates this vertex of the P([2, 2], 3) polytope from the other five is the following: X 11 + X 13 + (X 11 + X 21 ) + (X 12 + X 22 ) = 2X 11 + X 12 + X 13 + X 21 + X 22 = 3.5 Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

20 Sketch of the vertex proof Now using the entries of each matrix plugged into our equation: 2X 11 + X 12 + X 13 + X 21 + X 22 = Graph (a): X 11 = 1, X 12 = 1, X 13 =, X 21 =, X 22 =, X 23 = = 3; Graph (b): X 11 = 1, X 12 =, X 13 = 1, X 21 =, X 22 =, X 23 = = 3; Graph (c): X 11 =, X 12 = 1, X 13 = 1, X 21 = 1, X 22 =, X 23 = = 3; Graph (d): X 11 =, X 12 = 1, X 13 = 1, X 21 =, X 22 =, X 23 = = 2; Graph (e): X 11 = 1, X 12 =, X 13 = 1, X 21 =, X 22 = 1, X 23 = = 4; Graph (f): X 11 =, X 12 = 1, X 13 = 1, X 21 = 1, X 22 =-1, X 23 = (-1) = 2. You can see that our vertex is on one side of 2X 11 + X 12 + X 13 + X 21 + X 22 = 3.5 and the other five are on the other. Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

21 Inequality description of P(λ, n) Theorem (S. and Striker) P(λ, n) consists of all λ 1 n real matrices X = (X ij ) such that: i X ij 1 for all 1 i λ 1, 1 j n i=1 j X ij for all 1 j n, 1 i λ 1 j=1 n X ij = l i l i 1 for all 1 i λ 1 j=1 Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

22 Sketch of the proof of the inequality description Left: A matrix in P([3, 3, 1], 4); Right: An open circuit graph Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

23 Sketch of the proof of the inequality description = The decomposition of the previous matrix. Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

24 Enumeration of P(λ, n) Definition A facet is a face of the polytope that is dimension one less than the polytope. Conjecture The formula for the number of facets of P(λ, n) for rectangular and square tableaux is 3nλ 1 n 5λ Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

25 Main Topics 1 Background 2 Polytope with shape λ and max entry n 3 Polytope of m n sign matrices 4 Comparing P(λ, n) and P(m, n) 5 Future Connections Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

26 P(m, n) Polytope Definition Let P(m, n) be the polytope defined as the convex hull of all m n sign matrices. Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

27 P(m, n) Polytope Definition Let P(m, n) be the polytope defined as the convex hull of all m n sign matrices. Theorem (S. and Striker) The vertices of P(m, n) are the sign matrices of size m n. Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

28 Inequality description of P(m, n) Theorem (S. and Striker) P(m, n) consists of all m n real matrices X = {x ij } such that: i x ij 1 for all 1 i m, 1 j n. i=1 j x ij for all 1 i m, 1 j n. j=1 Thus all of the partial sums of the columns are between and 1 and the partial sums of all the rows are nonnegative. Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

29 Enumerations Theorem (S. and Striker) The dimension of P(m, n) is mn for all m > 1. Theorem (S. and Striker) There are n(3m 1) 2(m 2) facets in the P(m, n) polytope. Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

30 Main Topics 1 Background 2 Polytope with shape λ and max entry n 3 Polytope of m n sign matrices 4 Comparing P(λ, n) and P(m, n) 5 Future Connections Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

31 Compare P(λ, n) and P(m, n) What is different between P(λ, n) and P(m, n)? Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

32 Compare P(λ, n) and P(m, n) What is different between P(λ, n) and P(m, n)? P(λ, n) has a fixed shape of tableau which give a sign matrix of size λ 1 n. Then these certain sign matrices are used for the convex hull of P(λ, n). P(m, n) is the convex hull of all m n sign matrices. They correspond to all tableaux that fit into an m n box. Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

33 Compare P(λ, n) and P(m, n) (, 1, 1) (1, 1, 1) (, 1, ) (1, 1, ) (,, 1) (1,, 1) (,, ) P(1, 3) (1,, ) Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

34 Compare P(λ, n) and P(m, n) (, 1, 1) (1, 1, 1) (, 1, ) (1, 1, ) 2 1 (,, 1) 2 3 (1,, 1) 3 1 (,, ) (1,, ) 1 P(1, 3) and the corresponding tableaux. Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

35 Compare P(λ, n) and P(m, n) P ([1, 1], 3) P ([1, 1, 1], 3) P ([1], 3) (, 1, ) 2 P ([ ], 3) (,, ) (, 1, 1) 2 3 (,, 1) 3 (1, 1, 1) (1, 1, ) 1 2 (1,, 1) 3 1 (1,, ) 1 P(1, 3) and the corresponding tableaux. Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

36 Main Topics 1 Background 2 Polytope with shape λ and max entry n 3 Polytope of m n sign matrices 4 Comparing P(λ, n) and P(m, n) 5 Future Connections Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

37 Comparing the ASM and Birkhoff Polytopes Birkhoff ASMn P(m, n) Dimension (n 1) 2 (n 1) 2 mn Inequality rows and columns sum to 1 partial columns 1 Description entries partial sums row partials Vertices n! n 1 j= (3j + 1)! (n + j)! 1 i j n m + i + j 1 i + j 1 Facets n 2 4[(n 2) 2 + 1] 3mn n 2(m 1) Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

38 What is next? Are there any connections to these polytopes? Transportation Polytopes Gelfand-Tsetlin Polytope Other things with these polytopes? Describe the face lattice Explore more with P(m, n) vs. P(n, m) vs. P(λ, n). Look into the symmetry and other possible geometry Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37

39 Thanks! Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, / 37