Character formulas and matrices Ron Adin and Yuval Roichman - PowerPoint PPT Presentation

Character formulas and matrices Ron Adin and Yuval Roichman Department of Mathematics Bar-Ilan University   1 1 1 1 1 − 1 1 − 1     1 1 − 1 − 1   − 1 1 0 1

1. Character formulas 2. Matrices 3. Back to characters Abstract We present a family of square matrices which are asymmetric variants of Walsh-Hadamard matrices. They originate in the study of character formulas, and provide a handy tool for translation of statements about permutation statistics to results in representation theory, and vice versa. They turn out to have many fascinating properties.

1. Character formulas 2. Matrices 3. Back to characters Outline 1. Character formulas 2. Matrices 3. Back to characters

1. Character formulas 2. Matrices 3. Back to characters Character formulas

1. Character formulas 2. Matrices 3. Back to characters µ -unimodal permutations • A sequence ( a 1 , . . . , a n ) of distinct positive integers is unimodal if there exists 1 ≤ m ≤ n such that a 1 > a 2 > . . . > a m < a m +1 < . . . < a n .

1. Character formulas 2. Matrices 3. Back to characters µ -unimodal permutations • A sequence ( a 1 , . . . , a n ) of distinct positive integers is unimodal if there exists 1 ≤ m ≤ n such that a 1 > a 2 > . . . > a m < a m +1 < . . . < a n . • Let µ = ( µ 1 , . . . , µ t ) be a composition of n . A sequence of n positive integers is µ -unimodal if the first µ 1 integers form a unimodal sequence, the next µ 2 integers form a unimodal sequence, and so on.

1. Character formulas 2. Matrices 3. Back to characters µ -unimodal permutations • A sequence ( a 1 , . . . , a n ) of distinct positive integers is unimodal if there exists 1 ≤ m ≤ n such that a 1 > a 2 > . . . > a m < a m +1 < . . . < a n . • Let µ = ( µ 1 , . . . , µ t ) be a composition of n . A sequence of n positive integers is µ -unimodal if the first µ 1 integers form a unimodal sequence, the next µ 2 integers form a unimodal sequence, and so on. • A permutation π ∈ S n is µ -unimodal if the sequence ( π (1) , . . . , π ( n )) is µ -unimodal.

1. Character formulas 2. Matrices 3. Back to characters µ -unimodal permutations, descent set • Let U µ be the set of all µ -unimodal permutations in S n .

1. Character formulas 2. Matrices 3. Back to characters µ -unimodal permutations, descent set • Let U µ be the set of all µ -unimodal permutations in S n . • Example: n = 10, µ = (3 , 3 , 4). π = (4 , 2 , 10 , 9 , 7 , 6 , 5 , 3 , 1 , 8) ∈ U µ | | µ 2 | | µ 1 µ 3

1. Character formulas 2. Matrices 3. Back to characters µ -unimodal permutations, descent set • Let U µ be the set of all µ -unimodal permutations in S n . • Example: n = 10, µ = (3 , 3 , 4). π = (4 , 2 , 10 , 9 , 7 , 6 , 5 , 3 , 1 , 8) ∈ U µ | | µ 2 | | µ 1 µ 3 • The descent set of a permutation π ∈ S n is Des( π ) := { i : π ( i ) > π ( i + 1) } .

1. Character formulas 2. Matrices 3. Back to characters µ -unimodal permutations, descent set • Let U µ be the set of all µ -unimodal permutations in S n . • Example: n = 10, µ = (3 , 3 , 4). π = (4 , 2 , 10 , 9 , 7 , 6 , 5 , 3 , 1 , 8) ∈ U µ | | µ 2 | | µ 1 µ 3 • The descent set of a permutation π ∈ S n is Des( π ) := { i : π ( i ) > π ( i + 1) } . • Example: Des( π ) = { 1 , 3 , 4 , 5 , 6 , 7 , 8 }

1. Character formulas 2. Matrices 3. Back to characters µ -unimodal permutations, descent set • Let U µ be the set of all µ -unimodal permutations in S n . • Example: n = 10, µ = (3 , 3 , 4). π = (4 , 2 , 10 , 9 , 7 , 6 , 5 , 3 , 1 , 8) ∈ U µ | | µ 2 | | µ 1 µ 3 • The descent set of a permutation π ∈ S n is Des( π ) := { i : π ( i ) > π ( i + 1) } . • Example: Des( π ) = { 1 , 3 , 4 , 5 , 6 , 7 , 8 } • Denote I ( µ ) := { 1 , . . . , n } \ { µ 1 , µ 1 + µ 2 , µ 1 + µ 2 + µ 3 , . . . }

1. Character formulas 2. Matrices 3. Back to characters µ -unimodal permutations, descent set • Let U µ be the set of all µ -unimodal permutations in S n . • Example: n = 10, µ = (3 , 3 , 4). π = (4 , 2 , 10 , 9 , 7 , 6 , 5 , 3 , 1 , 8) ∈ U µ | | µ 2 | | µ 1 µ 3 • The descent set of a permutation π ∈ S n is Des( π ) := { i : π ( i ) > π ( i + 1) } . • Example: Des( π ) = { 1 , 3 , 4 , 5 , 6 , 7 , 8 } • Denote I ( µ ) := { 1 , . . . , n } \ { µ 1 , µ 1 + µ 2 , µ 1 + µ 2 + µ 3 , . . . } • Example: I ( µ ) = { 1 , . . . , 10 } \ { 3 , 6 , 10 } = { 1 , 2 , 4 , 5 , 7 , 8 , 9 } Des( π ) ∩ I ( µ ) = { 1 , 4 , 5 , 7 , 8 }

1. Character formulas 2. Matrices 3. Back to characters Formula 1: irreducible characters Let λ and µ be partitions of n , let χ λ be the character of the irreducible S n -representation corresponding to λ , and let χ λ µ be its value on a conjugacy class of cycle type µ . Theorem (Roichman ’97) � χ λ ( − 1) | Des( π ) ∩ I ( µ ) | , µ = π ∈C∩ U µ where C is any Knuth class of shape λ .

1. Character formulas 2. Matrices 3. Back to characters Formula 2: coinvariant algebra, homogeneous component Let χ ( k ) be the S n -character corresponding to the symmetric group action on the k -th homogeneous component of its coinvariant algebra, and let χ ( k ) be its value on a conjugacy class of cycle type µ µ . Theorem (A-Postnikov-Roichman, ’00) χ ( k ) � ( − 1) | Des( π ) ∩ I ( µ ) | , = µ π ∈ L ( k ) ∩ U µ where L ( k ) is the set of all permutations of length k in S n .

1. Character formulas 2. Matrices 3. Back to characters Formula 3: Gelfand model A complex representation of a group or an algebra A is called a Gelfand model for A if it is equivalent to the multiplicity free direct sum of all irreducible A -representations. Let χ G be the corresponding character, and let χ G µ be its value on a conjugacy class of cycle type µ . Theorem (A-Postnikov-Roichman, ’08) The character of the Gelfand model of S n at a conjugacy class of cycle type µ is equal to � χ G ( − 1) | Des( π ) ∩ I ( µ ) | , µ = π ∈ Inv n ∩ U µ where Inv n := { σ ∈ S n : σ 2 = id } is the set of all involutions in S n .

1. Character formulas 2. Matrices 3. Back to characters Inverse formulas? Question Are these formulas invertible? In other words: to what extent do the character values χ ∗ µ ( ∀ µ ) determine the distribution of descent sets?

1. Character formulas 2. Matrices 3. Back to characters Matrices

1. Character formulas 2. Matrices 3. Back to characters Subsets as indices Definition Let P n be the power set (set of all subsets) of { 1 , . . . , n } , with the anti-lexicographic linear order: for I , J ∈ P n , I � = J , let m be the largest element in the symmetric difference I △ J := ( I ∪ J ) \ ( I ∩ J ), and define: I < J ⇐ ⇒ m ∈ J . Example The linear order on P 3 is ∅ < { 1 } < { 2 } < { 1 , 2 } < { 3 } < { 1 , 3 } < { 2 , 3 } < { 1 , 2 , 3 } . P n will index the rows and columns of our matrices.

1. Character formulas 2. Matrices 3. Back to characters Walsh-Hadamard matrices The Walsh-Hadamard matrix H n of order 2 n has entries h I , J := ( − 1) | I ∩ J | ( ∀ I , J ∈ P n ) .

1. Character formulas 2. Matrices 3. Back to characters Walsh-Hadamard matrices The Walsh-Hadamard matrix H n of order 2 n has entries h I , J := ( − 1) | I ∩ J | ( ∀ I , J ∈ P n ) . Example � 1 � 1 H 1 = 1 − 1

1. Character formulas 2. Matrices 3. Back to characters Walsh-Hadamard matrices The Walsh-Hadamard matrix H n of order 2 n has entries h I , J := ( − 1) | I ∩ J | ( ∀ I , J ∈ P n ) . Example � 1 � 1 H 1 = 1 − 1   1 1 1 1 − 1 − 1 1 1  = H ⊗ 2   H 2 =   1 1 1 − 1 − 1  − 1 1 1 1

1. Character formulas 2. Matrices 3. Back to characters Walsh-Hadamard matrices The Walsh-Hadamard matrix H n of order 2 n has entries h I , J := ( − 1) | I ∩ J | ( ∀ I , J ∈ P n ) . Example � 1 � 1 H 1 = 1 − 1   1 1 1 1 − 1 − 1 1 1  = H ⊗ 2   H 2 =   1 1 1 − 1 − 1  − 1 1 1 1 H t H n H t n = 2 n I 2 n n = H n

1. Character formulas 2. Matrices 3. Back to characters Prefixes and runs Definition The prefix of length p of an interval { m + 1 , . . . , m + ℓ } is the interval { m + 1 , . . . , m + p } (0 ≤ p ≤ ℓ ).

1. Character formulas 2. Matrices 3. Back to characters Prefixes and runs Definition The prefix of length p of an interval { m + 1 , . . . , m + ℓ } is the interval { m + 1 , . . . , m + p } (0 ≤ p ≤ ℓ ). Definition For I ∈ P n let I 1 , . . . , I t be the sequence of runs (maximal consecutive intervals) in I .

1. Character formulas 2. Matrices 3. Back to characters Prefixes and runs Definition The prefix of length p of an interval { m + 1 , . . . , m + ℓ } is the interval { m + 1 , . . . , m + p } (0 ≤ p ≤ ℓ ). Definition For I ∈ P n let I 1 , . . . , I t be the sequence of runs (maximal consecutive intervals) in I . Example For I = { 1 , 2 , 4 , 5 , 6 , 8 , 10 } ∈ P 10 : I 1 = { 1 , 2 } , I 2 = { 4 , 5 , 6 } , I 3 = { 8 } , I 4 = { 10 } .

1. Character formulas 2. Matrices 3. Back to characters The matrices A and B