A Sundaram type bijection for SO (2 k + 1): vacillating tableaux and - PowerPoint PPT Presentation

A Sundaram type bijection for SO (2 k + 1): vacillating tableaux and pairs consisting of a standard Young tableau and an orthogonal Littlewood-Richardson tableau Judith Jagenteufel 31 st Conference on Formal Power Series and Algebraic Combinatorics (FPSAC), Ljubljana, July 2 nd , 2019 1 3 1 1 � � , 2 4 2 2 5 1 Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

Classical Schur-Weyl duality V ⊗ r ∼ � V GL ( λ ) ⊗ S ( λ ) = λ ⊢ r ℓ ( λ ) ≤ n as GL ( V ) × S r representations, V = C n . ◮ GL ( V ) acts diagonally and S r permutes tensor positions ◮ V GL ( λ ), S ( λ ) . . . irreducible representation of GL ( V ), S r Robinson-Schensted { 1 , . . . , n } r ↔ � � � SSYT( λ ) , SYT( λ ) λ ◮ SSYT( λ ), SYT( λ ) . . . (semi)standard Young tableaux Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

Special orthogonal group branching rule V GL ( λ ) ↓ GL ( V ) SO ( V ) ∼ � c µ λ V SO ( µ ) = µ where c µ λ are multiplicities counted by orthogonal LR tableaux leads to V ⊗ r ∼ � � c µ � V SO ( µ ) ⊗ V SO ( µ ) ⊗ U ( r , µ ) = λ S ( λ ) = µ a partition λ ⊢ r µ a partition l ( λ ) ≤ n l ( µ ) ≤ n l ( µ ) ≤ n µ ′ 1 + µ ′ µ ′ 1 + µ ′ 2 ≤ n 2 ≤ n as SO ( n ) × S r representations. n = 2 k + 1 thus n odd ◮ V = C n . . . vector representation of SO ( n ) ◮ V SO ( µ ), S ( λ ) . . . irreducible representations of SO ( n ) and S r Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

Our setting � � c µ � U ( r , µ ) = λ S ( λ ) λ Our main result: a bijection between � � vacillating tableaux ↔ orthogonal LRT, SYT that preserves descents. Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

Standard Young tableaux 1 2 5 6 9 3 4 7 8 12 1013 Standard Young tableaux of shape λ ( SYT ( λ )): 1116 14 fillings of a Young diagram of shape λ with entries 15 { 1 , 2 , . . . , | λ |} , increasing in rows and columns 17 1 2 5 6 9 Descents 3 4 7 8 12 d is a descent if d + 1 is in a row below d 1013 1116 14 15 17 descent set { 2 , 6 , 9 , 10 , 12 , 13 , 14 , 16 } Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

Vacillating tableaux - highest weight words ◮ Vacillating tableau : a sequence of partitions / Young diagrams ∅ = λ 0 , λ 1 , . . . , λ r , at most k rows, shape λ r ◮ λ i and λ i +1 differ in at most one cell ◮ λ i = λ i +1 only if k th row is non-empty ◮ Highest weight word : word w with letters in { 0 , ± 1 , . . . , ± k } , length r , weight (#1 − #( − 1) , . . . , # k − #( − k )), such that for every prefix w 1 , . . . , w j : ◮ # i − #( − i ) ≥ 0 ◮ # i − #( − i ) ≥ #( i + 1) − #( − i − 1) ◮ If the last position, w j = 0 then # k − #( − k ) > 0. Example ( k = 3) ( 1 , 3 , − 3 , − 1 , − 2 , − 1) 1 , 2 , − 2 , 1 , 1 , 2 , 2 , 1 , 3 , 0 , 2 , ∅ Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

Descents of vacillating tableaux Vacillating tableaux a position i of w is a descent if there exists a path from w i to w i +1 in the crystal graph: 1 → 2 → · · · → k → 0 → − k → · · · → − 1 and w i w i +1 � = j ( − j ) if # j − #( − j ) = 0 in w 1 , . . . , w i − 1 . Example 6 7 8 9 1011121314151617 2 3 4 5 1011121314 16 1 8 3 4 7 1314 1011 descent set { 2 , 6 , 9 , 10 , 12 , 13 , 14 , 16 } Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

Quasi symmetric expansion Recall the Frobenius character: ch ρ = 1 � Tr ρ ( π ) p λ ( π ) r ! π ∈ S r where p λ is a power sum symmetric function. We have: � ch S ( λ ) = s λ = F Des ( Q ) Q ∈ SYT ( λ ) where s λ is a Schur function and F D is a fundamental quasi-symmetric function: � F D = x i 1 x i 2 . . . x i r . i 1 ≤ i 2 ≤···≤ i r j ∈ D ⇒ i j < i j +1 Therefore, as our bijection is descent preserving, we obtain: � � c µ � � λ S ( λ ) = ch F Des ( w ) . λ ⊢ r w vacillating tableau l ( λ ) ≤ n of length r and shape µ Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

Main result: a bijection between � � vacillating tableaux ↔ orthogonal LRT, SYT that preserves descents. Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

Strategy (SYT, oLRT) (SYT, aoLRT) (SYT odd, µ ) 1 2 5 6 9 1 1 1 2 5 6 9 1 2 5 6 9 3 4 7 812 2 2 3 4 7 812 3 4 7 812 1013 1 1 3 3 1013 3 2 1 1013 18 20 23 1116 2 2 4 1116 2 1116 19 14 1 1 1 14 1 1 14 2122 15 4 5 15 15 µ = (3 , 2 , 1) 17 5 17 17 Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

Orthogonal Littlewood-Richardson tableaux n = 2 k + 1, ℓ ( λ ) ≤ n , ℓ ( µ ) ≤ k , µ ≤ λ e.g. n = 7, k = 3, λ = (5 , 5 , 2 , 2 , 1 , 1 , 1), µ = (3 , 2 , 1) Kwon’s LR tableaux alternative LR tableaux ◮ reverse skew-shape ◮ k twocolumn skew-shape semistandard tableau with semistandard tableaux, tail µ i ; one single column inner shape λ ◮ λ ′ determines the filling ◮ µ determines the filling, reading word is Yamanouchi ◮ several conditions on size ◮ technical condition and filling 1 1 2 2 3 2 1 1 1 3 3 2 2 2 4 1 1 1 1 1 4 5 5 Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

First part: manipulate the orthogonal LR tableaux 3 1 1 3 2 2 �→ �→ 1 1 3 3 2 2 4 1 1 1 4 5 5 3 3 3 2 �→ �→ �→ 2 2 2 2 2 �→ �→ �→ �→ 2 3 2 3 2 3 2 3 2 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

Strategy (SYT, oLRT) (SYT, aoLRT) (SYT odd, µ ) 1 2 5 6 9 1 1 1 2 5 6 9 1 2 5 6 9 3 4 7 812 2 2 3 4 7 812 3 4 7 812 1013 1 1 3 3 1013 3 2 1 1013 18 20 23 1116 2 2 4 1116 2 1116 19 14 1 1 1 14 1 1 14 2122 15 4 5 15 15 µ = (3 , 2 , 1) 17 5 17 17 (SYT even, µ ) (vac. tab. even, shape ∅ , partition) 1 8 9121316 2 1011141519 µ = (3 , 2 , 1) 3 1720 25 27 30 4 1823 26 5 21 2829 6 22 µ = (3 , 2 , 1) 7 24 (vac. tab. odd, shape ∅ , µ ) vacillating tableau µ = (3 , 2 , 1) Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)

1 8 9121316 2 1011141519 3 1720 25 27 30 16 17 14 15 19 4 1823 26 13 20 5 21 2829 12 25 910 11 1 2 8 910 13 14 27 6 22 11 12 15 16 1 2 3 8 1 8 912 13 16 19 30 7 24 2 10 11 14 15 19 17 17 18 19 20 16 18 16 21 14 15 19 20 14 15 13 23 13 23 25 25 26 27 12 26 12 28 910 11 910 11 27 29 1 2 3 4 8 1 2 3 4 5 8 15 17 20 25 15 17 19 20 30 18 19 26 30 10 12 14 2 25 2 4 10 12 14 27 17 20 25 3 3 3 17 18 19 20 21 16 22 14 15 13 23 25 26 27 12 28 910 11 20 19 21 29 1 2 3 4 5 6 8 15 17 18 25 26 30 4 5 2 10 11 14 27 17 18 19 20 21 16 22 17 18 20 25 14 15 23 3 13 24 25 26 27 3 4 12 28 910 11 20 21 23 25 19 29 1 2 3 4 5 6 7 8 15 17 18 26 30 4 2 6 10 11 14 19 20 21 27 18 25 3 5 3 4 5 Judith Jagenteufel A Sundaram type bijection for SO (2 k + 1)