The Asymptotic Distribution of Symbols on Staircase Tableaux - PowerPoint PPT Presentation

Definitions Connections/Motivation Previous Result Results The Asymptotic Distribution of Symbols on Staircase Tableaux Diagonals Amanda Lohss September 15, 2016 Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Outline Definition of Staircase Tableaux Connections/Motivation Previous Results Results Open Problems Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Staircase Tableaux (Corteel-Williams (2010)) Definition α γ α A staircase tableau of size n is a δ Young diagram of shape β γ (n, n-1, ..., 1) such that: δ 1 The boxes are empty or contain β an α , β , γ , or δ . γ β Figure: An example of a staircase tableau of size 7. Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Staircase Tableaux (Corteel-Williams (2010)) Definition α γ α A staircase tableau of size n is a δ Young diagram of shape β γ (n, n-1, ..., 1) such that: δ 1 The boxes are empty or contain β an α , β , γ , or δ . γ 2 Every box on the diagonal β contains a symbol. Figure: An example of a staircase tableau of size 7. Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Staircase Tableaux (Corteel-Williams (2010)) Definition α γ α A staircase tableau of size n is a δ Young diagram of shape β γ (n, n-1, ..., 1) such that: δ 1 The boxes are empty or contain β an α , β , γ , or δ . γ 2 Every box on the diagonal β contains a symbol. 3 All boxes in the same column Figure: An example of and above an α or γ are empty. a staircase tableau of size 7. Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Staircase Tableaux (Corteel-Williams (2010)) Definition α γ α A staircase tableau of size n is a δ Young diagram of shape β γ (n, n-1, ..., 1) such that: δ 1 The boxes are empty or contain β an α , β , γ , or δ . γ 2 Every box on the diagonal β contains a symbol. 3 All boxes in the same column Figure: An example of and above an α or γ are empty. a staircase tableau of 4 All boxes in the same row and to size 7. the left of an β or δ are empty. Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Preliminaries The rows and columns in a α γ α staircase tableau are numbered δ from 1 through n , beginning β γ with the box in the NW-corner δ and continuing south and east β respectively. γ β Figure: A staircase tableau with weight α 2 β 3 γ 3 δ 2 . Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Preliminaries The rows and columns in a α γ α staircase tableau are numbered δ from 1 through n , beginning β γ with the box in the NW-corner δ and continuing south and east β respectively. γ Symmetric with respect to β interchanging rows/columns, α / β , and γ / δ . Figure: A staircase tableau with weight α 2 β 3 γ 3 δ 2 . Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Preliminaries The rows and columns in a α γ α staircase tableau are numbered δ from 1 through n , beginning β γ with the box in the NW-corner δ and continuing south and east β respectively. γ Symmetric with respect to β interchanging rows/columns, α / β , and γ / δ . Figure: A staircase The weight of a staircase tableau with weight tableau is the product of all its α 2 β 3 γ 3 δ 2 . symbols. Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Preliminaries Cont’d As proven by Corteel and Williams, α γ α summing over the weight of all staircase δ tableaux gives, β γ δ � wt ( S ) = β S ∈S n γ n − 1 β � ( α + β + δ + γ + i ( α + γ )( β + δ )) . i =0 Figure: A staircase and therefore the total number of staircase tableau with weight tableaux is 4 n · n !. α 2 β 3 γ 3 δ 2 . Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Connections Introduced due to connections with the asymmetric simple exclusion process (ASEP), an important particle model with applications in physics, biology and biochemistry. Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Connections Introduced due to connections with the asymmetric simple exclusion process (ASEP), an important particle model with applications in physics, biology and biochemistry. According to Yau, the ASEP is “the default stochastic model for transport phenomena.” Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Connections Introduced due to connections with the asymmetric simple exclusion process (ASEP), an important particle model with applications in physics, biology and biochemistry. According to Yau, the ASEP is “the default stochastic model for transport phenomena.” Numerous other connections such as Askey-Wilson polynomials, tree–like tableaux, permutation tableaux, and permutations. Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results The ASEP A Markov Chain with n sites. ◦ ◦ • • ◦ • ◦ ◦ • Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results The ASEP A Markov Chain with n sites. ◦ ◦ • • ◦ • ◦ ◦ • Transition Probabilities: α γ ◦ A to • A : • A to ◦ A : n + 1 n + 1 Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results The ASEP A Markov Chain with n sites. ◦ ◦ • • ◦ • ◦ ◦ • Transition Probabilities: α γ ◦ A to • A : • A to ◦ A : n + 1 n + 1 δ β A ◦ to A • : A • to A ◦ : n + 1 n + 1 Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results The ASEP A Markov Chain with n sites. ◦ ◦ • • ◦ • ◦ ◦ • Transition Probabilities: α γ ◦ A to • A : • A to ◦ A : n + 1 n + 1 δ β A ◦ to A • : A • to A ◦ : n + 1 n + 1 u q A • ◦ B to A ◦ • B : A ◦ • B to A • ◦ B : n + 1 n + 1 Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Connection with the ASEP Type of a staircase tableaux: • for each α or δ on diagonal. ◦ for each β or γ on diagonal. α γ α δ β γ δ β γ β Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Connection with the ASEP Type of a staircase tableaux: • for each α or δ on diagonal. ◦ for each β or γ on diagonal. α γ α • δ • β γ ◦ δ • β ◦ γ ◦ β ◦ Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Connection with the ASEP Type of a staircase tableaux: • for each α or δ on diagonal. ◦ for each β or γ on diagonal. α γ α • δ Filling rules for u ’s and q ’s: • β γ ◦ 1 u ’s in all boxes east of a β and δ • q ’s in all boxes east of a δ . β ◦ γ ◦ 2 β ◦ Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Connection with the ASEP Type of a staircase tableaux: • for each α or δ on diagonal. ◦ for each β or γ on diagonal. α γ α • δ Filling rules for u ’s and q ’s: • u u β γ ◦ 1 u ’s in all boxes east of a β and δ • q ’s in all boxes east of a δ . β u u ◦ γ ◦ 2 β ◦ Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Connection with the ASEP Type of a staircase tableaux: • for each α or δ on diagonal. ◦ for each β or γ on diagonal. α γ α • δ Filling rules for u ’s and q ’s: q q q q q • u u β γ ◦ 1 u ’s in all boxes east of a β and q q q δ • q ’s in all boxes east of a δ . β u u ◦ γ ◦ 2 β ◦ Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Connection with the ASEP Type of a staircase tableaux: • for each α or δ on diagonal. ◦ for each β or γ on diagonal. α γ α • δ Filling rules for u ’s and q ’s: q q q q q • u u β γ ◦ 1 u ’s in all boxes east of a β and q q q δ • q ’s in all boxes east of a δ . β u u ◦ γ ◦ 2 u ’s in all boxes north of a α or δ β ◦ and q ’s in all boxes north of a β or γ . Amanda Lohss Purdue 2016

Definitions Connections/Motivation Previous Result Results Connection with the ASEP Type of a staircase tableaux: • for each α or δ on diagonal. ◦ for each β or γ on diagonal. α γ u α • δ Filling rules for u ’s and q ’s: q q q q q • u u β u γ ◦ 1 u ’s in all boxes east of a β and q q q δ • q ’s in all boxes east of a δ . β u u ◦ γ ◦ 2 u ’s in all boxes north of a α or δ β ◦ and q ’s in all boxes north of a β or γ . Amanda Lohss Purdue 2016