Descent sets for oscillating tableaux Martin Rubey 1 Bruce Sagan 2 - PowerPoint PPT Presentation

Descent sets for oscillating tableaux Martin Rubey 1 Bruce Sagan 2 Bruce Westbury 3 1 TU Wien 2 Michigan State University 3 University of Warwick

n -symplectic oscillating tableaux µ 0 µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 = µ ∅ an oscillating tableau is a sequence of partitions ( µ 0 , µ 1 , . . . , µ r ) ◮ beginning with ∅ ◮ Ferrers diagrams of consecutive partitions differ by precisely one cell

n -symplectic oscillating tableaux µ 0 µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 = µ ∅ an oscillating tableau is a sequence of partitions ( µ 0 , µ 1 , . . . , µ r ) ◮ beginning with ∅ ◮ Ferrers diagrams of consecutive partitions differ by precisely one cell

n -symplectic oscillating tableaux µ 0 µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 = µ ∅ an oscillating tableau is a sequence of partitions ( µ 0 , µ 1 , . . . , µ r ) ◮ beginning with ∅ ◮ Ferrers diagrams of consecutive partitions differ by precisely one cell ◮ r is the length r = 9 ◮ µ = µ r is the (final) shape µ = ( 21 ) ◮ n -symplectic if µ i has at most n parts for all i n ≥ 3

n -symplectic oscillating tableaux µ 0 µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 = µ ∅ an oscillating tableau is a sequence of partitions ( µ 0 , µ 1 , . . . , µ r ) ◮ beginning with ∅ ◮ Ferrers diagrams of consecutive partitions differ by precisely one cell ◮ r is the length r = 9 ◮ µ = µ r is the (final) shape µ = ( 21 ) ◮ n -symplectic if µ i has at most n parts for all i n ≥ 3 why are n -symplectic oscillating tableaux interesting?

combinatorialist’s answer n -symplectic oscillating tableaux of length r and empty shape and ( n + 1 ) -noncrossing perfect matchings of { 1 , 2 , . . . , r } are in bijection [Sundaram, Chen-Deng-Du-Stanley-Yan]! but that’s not for today. . .

Schur-Weyl duality let V be the defining representation of the general linear group GL ( n ) and consider its r -th tensor power V ⊗ r : ◮ GL ( n ) acts diagonally ◮ S r acts by permuting tensor positions then V ⊗ r ∼ � = V ( µ ) ⊗ S ( µ ) µ ⊢ r ℓ ( µ ) ≤ n as GL ( n ) × S r modules. ( V ( µ ) and S ( µ ) are the irreducible representations of GL ( n ) and S r corresponding to the partition µ )

Robinson-Schensted correspondence the combinatorial counterpart of V ⊗ r ∼ � = V ( µ ) ⊗ S ( µ ) µ ⊢ r ℓ ( µ ) ≤ n is the Robinson-Schensted correspondence { 1 , . . . , n } r ↔ � SSYT ( µ, n ) × SYT ( µ ) µ ⊢ r ℓ ( µ ) ≤ n ◮ V ( µ ) has a basis indexed by SSYT ( µ, n ) , semistandard Young tableaux of shape µ , entries in { 1 , . . . , n } ◮ S ( µ ) has a basis indexed by SYT ( µ ) , standard Young tableaux of shape µ

‘symplectic’ Schur-Weyl duality let V be the defining representation of the symplectic group Sp ( 2 n ) and consider its r -th tensor power V ⊗ r : ◮ Sp ( 2 n ) acts diagonally ◮ S r acts by permuting tensor positions then V ⊗ r ∼ � V Sp ( µ ) ⊗ U ( n , r , µ ) = ℓ ( µ ) ≤ n as Sp ( 2 n ) × S r modules. ( V Sp ( µ ) is the irreducible representations of Sp ( 2 n ) corresponding to the partition µ , U ( n , r , µ ) is the isotypic component of type µ , an S r module)

Berele’s correspondence a combinatorial counterpart of V ⊗ r ∼ � V Sp ( µ ) ⊗ U ( n , r , µ ) = ℓ ( µ ) ≤ n is Berele’s correspondence {± 1 , . . . , ± n } r ↔ � K ( µ, n ) × Osc ( n , r , µ ) ℓ ( µ ) ≤ n ◮ V Sp ( µ ) has a basis indexed by K ( µ, n ) , King’s n-symplectic semistandard tableaux of shape µ , entries in {± 1 , . . . , ± n } ◮ U ( n , r , µ ) has a basis indexed by Osc ( n , r , µ ) , n-symplectic oscillating tableaux of length r , shape µ

use n -symplectic oscillating tableaux to understand the isotypic components U ( n , r , µ ) ! in particular, compute their Frobenius character

Frobenius character the Frobenius map ch is a ring isomorphism between ◮ the ring of (virtual) characters of the symmetric group, and ◮ the ring of symmetric functions set ch U = ch χ for a representation U with character χ

Frobenius character the Frobenius map ch is a ring isomorphism between ◮ the ring of (virtual) characters of the symmetric group, and ◮ the ring of symmetric functions set ch U = ch χ for a representation U with character χ example let V be the defining representation of GL ( n ) by Schur-Weyl the isotypic component of type µ in V ⊗ r is S ( µ ) its Frobenius character is ch S ( µ ) = s µ

Sundaram’s correspondence to determine the Frobenius character of U ( n , r , µ ) , decompose it into S r -irreducibles: U ( n , r , µ ) ∼ � a ( λ, µ ) S ( λ ) = λ ⊢ r then � ch U ( n , r , µ ) = a ( λ, µ ) s λ λ ⊢ r

Sundaram’s correspondence the combinatorial counterpart of U ( n , r , µ ) ∼ � = a ( λ, µ ) S ( λ ) λ ⊢ r is Sundaram’s correspondence � Osc ( n , r , µ ) ↔ LR ( n , λ/µ, β ) × SYT ( λ ) λ ⊢ r β ⊢ r −| µ | β has even column lengths ◮ a ( λ, µ ) is the cardinality of LR ( n , λ/µ, β ) , the set of n -symplectic Littlewood-Richardson tableaux of shape λ/µ and weight β

the Frobenius character of U ( n , r , µ )   � �  c λ  ch U ( n , r , µ ) = µ,β ( n )  s λ    λ ⊢ r β ⊢ r −| µ | β has even column lengths where c λ µ,β ( n ) = # LR ( n , λ/µ, β )

the Frobenius character of U ( n , r , µ )   � �  c λ  ch U ( n , r , µ ) = µ,β ( n )  s λ    λ ⊢ r β ⊢ r −| µ | β has even column lengths where c λ µ,β ( n ) = # LR ( n , λ/µ, β ) we want something simpler!

quasisymmetric expansion the fundamental quasisymmetric functions are � x i 1 x i 2 · · · x i r . F D = i 1 ≤···≤ i r i j < i j + 1 if j ∈ D a descent in a standard Young tableau is an entry k such that k + 1 is in a lower row in English notation

quasisymmetric expansion the fundamental quasisymmetric functions are � x i 1 x i 2 · · · x i r . F D = i 1 ≤···≤ i r i j < i j + 1 if j ∈ D a descent in a standard Young tableau is an entry k such that k + 1 is in a higher row

quasisymmetric expansion the fundamental quasisymmetric functions are � x i 1 x i 2 · · · x i r . F D = i 1 ≤···≤ i r i j < i j + 1 if j ∈ D a descent in a standard Young tableau is an entry k such that k + 1 is in a higher row then, the Frobenius character of S ( µ ) can also be written as � ch S ( µ ) = s µ = F Des ( Q ) . Q ∈ SYT ( µ ) let’s do the same for the symplectic group

descents for oscillating tableaux µ 0 µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 = µ ∅

descents for oscillating tableaux µ 0 µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 = µ ∅

descents for oscillating tableaux µ 0 µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 = µ ∅ ¯ ¯ ¯ w : 1 2 1 2 1 1 2 3 3 ◮ convert the oscillating tableau to a highest weight word n < · · · < ¯ 2 < ¯ w 1 w 2 . . . w r with letters in 1 < 2 < · · · < n < ¯ 1

descents for oscillating tableaux ✓✏ ✓✏ ✓✏ ✓✏ ✓✏ ✓✏ µ 0 µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 = µ ✒✑ ✒✑ ✒✑ ✒✑ ✒✑ ✒✑ ∅ ¯ ¯ ¯ w : 1 2 1 2 1 1 2 3 3 ◮ convert the oscillating tableau to a highest weight word n < · · · < ¯ 2 < ¯ w 1 w 2 . . . w r with letters in 1 < 2 < · · · < n < ¯ 1 ◮ k is a descent if w k < w k + 1

quasisymmetric expansion Sundaram’s correspondence � Osc ( n , r , µ ) ↔ LR ( n , λ/µ, β ) × SYT ( λ ) λ ⊢ r β ⊢ r −| µ | β has even column lengths preserves descent sets: O ↔ ( L , Q ) ⇒ Des ( O ) = Des ( Q ) therefore � ch U ( n , r , µ ) = F Des ( O ) . O ∈ Osc ( n , r ,µ )

proof ∅ 1 2 21 11 1 11 21 31 21

proof ∅ ∅ 1 ∅ 1 2 ∅ 1 2 21 X ∅ 1 1 11 11 X ∅ ∅ ∅ 1 1 1 ∅ ∅ ∅ 1 1 1 11 ∅ ∅ ∅ 1 1 1 11 21 ∅ ∅ ∅ 1 1 1 11 21 31 X ∅ ∅ ∅ 1 1 1 11 21 21 21

proof ∅ X ∅ 1 X ∅ 1 2 ∅ 1 2 21 X ∅ 1 1 11 11 X ∅ ∅ ∅ 1 1 1 ∅ ∅ ∅ 1 1 1 11 ∅ ∅ ∅ 1 1 1 11 21 X ∅ ∅ ∅ 1 1 1 11 21 31 X ∅ ∅ ∅ 1 1 1 11 21 21 21 X X X

proof ∅ 1 2 21 31 41 411 421 431 441 X ∅ 1 2 21 31 31 311 321 331 431 X ∅ 1 2 21 21 21 211 221 321 421 ∅ 1 2 21 21 21 211 221 321 421 X ∅ 1 1 11 11 11 111 211 311 411 X ∅ ∅ ∅ 1 1 1 11 21 31 41 ∅ ∅ ∅ 1 1 1 11 21 31 41 ∅ ∅ ∅ 1 1 1 11 21 31 41 X ∅ ∅ ∅ 1 1 1 11 21 31 31 X ∅ ∅ ∅ 1 1 1 11 21 21 21 X X X

proof ∅ Q 1 2 21 31 41 411 421 431 441 X 1 X 2 21 X 11 X 1 11 21 X 31 X ∅ ∅ ∅ 1 1 1 11 21 21 21 X X X

proof ∅ Q 1 2 21 31 41 411 421 431 441 X 1 X 2 21 X 11 X 1 11 21 X 31 X ∅ ∅ ∅ 1 1 1 11 21 21 21 X X X