KRONECKER POWERS CHARACTER POLYNOMIALS. A. Goupil, with C. Chauve and A. Garsia,

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1 KRONECKER POWERS AND CHARACTER POLYNOMIALS A. Goupil, with C. Chauve and A. Garsia,

2 Menu Introduction : Kronecker products Tensor powers Character Polynomials Perspective : Duality with product of conjugacy classes 2

3 Tensor Product of representations of S n A : S n Aut(V ) a A( ) B : S n Aut(W ) a B( ) A B : S n Aut(V W ) a A( ) B( ) Definition. A B is called the Kronecker product of the representations A and B. When A and A µ are irreducible representations, then A A µ is, in general, not irreducible and A A µ = t,µ A The question of finding an easy computation and a combinatorial interpretation of the coefficients t,µ goes back to the beginning of representation theory. To compute the coefficients t,µ, we need the characters of the irreducible representations : t,µ = µ = C n! ( ) µ ( ) ( ) but we are looking for a combinatorial computation 3

4 - Tensor powers Expansion of the Kronecker powers of ( n,) : If P is the permutation representation then P = ( n,) ( n ) Notation : let ( n,)k = ( n,) ( n,) L ( n,) ( n,)k = t ( n,) k then we have the exponential generating function x k t ( n,) k k! = f x x! ee ( e x ) for all n kn + 2 k where = L Observe : t ( n,) k depends only on and f. We carry the Kronecker product in the ring of symmetric functions and we use the Schur functions basis. Ingredients needed : s : Schur functions s : operators adjoint to multiplication by s λ : Recall : s (s ) = s / = LR, s if γ λ 4

5 Combinatorial operator. (n,) µ = ( s () s * () )s µ (n,) k = ( s () s () ) k s (n ) * There exists such an operator on Schur functions for each µ s () s µ : remove one cell from the border of the diagram µ so that the remaining cells is a Ferrers diagram. s () s µ : add one cell to the border of the diagram µ so that the new set of cells is a Ferrers diagram. Example : ( s () s () )s (n ) : ( s () s () )s (n ) = ( s (n,) + s (n) ) s (n) = s (n-,) Apply this combinatorial operator k times and count 5

6 We obtain a walk in Young s Lattice (black edges) augmented with the red edges from (n) to λ : ( n,)k = number of walks of length k from (n) to λ. k > : oscillating and stationary tableaux of shape λ (S. Sundaram 986, L. Favreau al. 988 ) black edges : f k = Number of Oscillating Tableaux of shape λ and length k. k = 3 5 (k- λ -) ( f λ = card OT k 6

7 Idea of proof : : Oscillating tableaux: OT k bijection Right side: pairs, j < L < j ) ) 0.( r ( + ( +,Q + ( + : 2r + = k. - /. i, K, i r * * 2. (involution with no fixed point, standard tableau) on disjoint complementary subsets of [k]. + : Oscillating and stationary tableaux: bijection {( )} OST k,q - st component : π = set partition with parts of size nd component : Q λ = Standard tableau on a subset of [k]. each entry in Q λ is the largest entry in a part of π. - Entries in π and Q λ do not form disjoint sets. - Each number in [k] appears once or twice in a pair (π,q λ ) 7

8 OST k = f ) ( m =0 m = ( ( n,) ) k + (k*m ) / 2, m 2 ) ( p 2 (k * m,m 2 ) *m m 2 = *m where p 2 (a,b) = number of partitions of a set of size a in b parts of size 2. k ( ( n,) ) k y k k! = f y y ee ( e y ) Question: Can we extend these methods to other Kronecker powers? 8

9 2- Character Polynomials -Defined by Specht in 960, -Little known (Kerber gave a table) Definition. For each partition λ = λ λ 2,..., λ k of n and = λ 2,..., λ k, there is a (unique) polynomial q ( x,k,x n ), the Character Polynomial of λ, such that for all permutations σ Sn with cyclic type µ = m Ln m n, we have (n,) Examples. q λ (m,,m n ) = mln m n -Well known: µ (n,) = number of fixed points of Cµ q () (x) = x - = m- if µ = m 2 m 2 Ln m n 2- µ (n2,2) 3- µ (n2, 2 ) = m 2 + m ) ) 2 m ( q (2) (x) = x 2 + x 2 ( x = m 2 + m 2 ) ) ( q (,) (x) = -x 2 +(x -)(x -2)/2 9

10 Consequence of the definition : Products of character polynomials decompose precisely as Kronecker products : q q µ = t,µ q for all α, λ, µ n Example (suite). q ( x ) q ( x) (n,) (n,) = (n2,2) + (n2,2 ) + (n,) + (n) ( x ) 2. ( = x 2 + x +. ( 0 * - x 3 + x 2 + x + 0 * [ x ] + / ) 2, 2 / ) 2, 2 0

11 How to compute characters polynomials q λ (x)? Recipy. Using the umbral operators of Rota a) Write s λ in the power sums basis {p µ } µ n b) Replace pi by (ixi -) in each p µ = ( p ) m ( p 2 ) m 2 L c) Expand (ix i ) m i as a sum c x i i i d) Replace each x i i par x i i = (x i ) i (umbral operator) i Example. q(3)(x) a) s (3) = 6 (p 3 + 3p 2 + 2p 3 ) b) c) 6 ((x - ) 3 + 3(2x 2 - )(x - )+ 2(3x 3 - )) 6 (x 3 2-3x + 6x x 2-6x 2 + 6x 3 ) [ ] d) q ( 3) ( x ) = 6 (x ) 3-3(x ) 2 + 6x x 2-6x 2 + 6x 3

12 2 Recipy 2. Recursive Calculus à la Murnaghan-Nakayama a) Compute q ( x,0,...,0) = f ( x, ) as a polynomial in x. b) The term containing x i j and no variable x k, with k > i is x i j (() ht(s) q ) j ( x,...,x i(,0,...) S=() =) 0,),...,) j ), * where the sum is over all (j+)-tuples of partitions obtained by succesively removing from j border strips of length i. Example. q (3,,) (x, x 2, ) : f (x -5,3,,) + x 5-2 x 2 2 q ()(x) + x 2 q (,,)(x) - x 2 q (3)(x) = x! ( x 2)( x 5)( x 6)( x 8)!20 + x 5-2 x 2 2 ( x -) + x 2 x 3 ( ( x 3 ( ( x 2 ( ( ( ) * + +, -.. Proof. Symmetric functions, plethystic substitution

13 The Algebra of Character Polynomials The set { q ( x,x 2,K)} is a basis of Q[x,x 2, ] = Q[x] Scalar product in Q[x] : for f ( x,x 2,K,x k ), g( x,x 2,K,x k ), with i deg( x i ) k if f,g Q [ x ] = f ( a,ka k )g( a,ka k ), n 2k a Ln a n a! a 2!La n! = a 2 a 2Ln a n n then { q ( x )} becomes an orthonormal basis. We can use the scalar product, to compute Q [ x ] the expansion of any polynomial f Q[x] f = c q ( x ) c λ = f,q ( x) Q [ x ] Using truncated partitions, we can define the projective limit Z = of the centers Z n = n, of the group algebras of the symmetric groups S n. In summary, the application q : (Z, +, ) (Q[x], +, ), q(χ λ ) = q λ (x) is an algebra isomorphism and an isometry. i 3

14 Remark. We have also defined and computed character polynomials for Hecke Algebras of S n Example. µ (n,) = x - µ (n,) (q) = q nl( µ) x + (l(µ ) )(q ) [ ]. 4

15 - Coverings of {,2,, n} Bell numbers B k = ( h h n ) k,h n = APPLICATIONS k m z µ µ= m 2 m 2Lk m k k = number of walks of length k obtained from the action of s () s () on Ferrers diagrams starting and ending at Identity. h 2 : Multigraphs (G. Labelle) w * k = number of multigraphs with k labelled edges and no loop = ( h 2 h n2 ) k,h n = / µ= m 2 m 2Lk m k.2k ( m 2 + m + * - ) 2, = number of walks of length k obtained from the action of h 2 =s ( 2) s ( 2) + s (,) s (,) on Ferrers diagrams starting and ending at Identity. z µ k 5

16 h 3 : Russian dolls (D. Zeilberger,) number of coverings with k labelled triangles = ( h 3 h n3 ) k ) 3,h n = + ( z µ * + i= µ= m 2 m 2Lk m k 0 3k /= n 2 n 2Lr n r 0 3 = number of walks of length k obtained from the action of h 3 on Ferrers diagrams. m i n i,. -. k 2- Permutations The character polynomial k k q k ( n)= ( e ) ) (() r n( n( )L( n( k +) r=0 r = number of permutations S n with longest increasing subsequence of size n-k present at the beginning of σ : σ() < σ(2) < < σ(n-k)=n. 6

17 Class Polynomials Kronecker product is «dual» to product of conjugacy classes : Analogies. - Stability. 2- There exists a family {ω µ } µ of symmetric polynomials with ordinary product analogous to product of conjugacy classes Example. C( n-2,2)*c( n-3,3) = 4C( n-4,4)+c( n-5,3,2)+2(n-2)c( n-2,2) First examples known to Frobenius, 90 and Ingram, 950. ω ( n-2, 2 ) = p (x) ω ( n-3, 3 ) = p 2 (x) - n 2 ω ( n-4, 4 ) = p 3 (x) (2n-3)p (x) ω ( n-5, 3,2 ) = p (2,) (x) 4p 3 (x) ( n 2-6n+8)p (x) ω ( n-2,2) ω (n-3,3) = 4 ω (n-4,4) + ω (n-5,3,2) + 2(n-2) ω (n-2,2) 7

18 Fundamental Property of the ω µ (x) : The value of ω µ (x) on the contents C(λ)={(j-i)} (i,j) λ of a Ferrers diagram λ, is ω µ (C(λ)) = C µ( n ) f µ 8

19 Class polynomials w µ (x) vs Character polynomials p µ (x) Can we establish a combinatorial link between Kronecker coefficients and Class coefficients? 9

20 References A. Goupil, C. Chauve, Combinatorial operators for Kronecker powers of representations of Sn, Séminaire lotharingien de combinatoire, no 54, A. Garsia, A.Goupil, Character Polyominals, their q-analogs and the Kronecker product, The Electronic Journal of Combinatorics, no 6 (2),