Báo cáo toán học: " Character Polynomials, their q-analogs and the Kronecker product" potx

Character Polynomials, their q-analogs and the Kronecker product A. M. Garsia* and A. Goupil ** Department of Mathematics University of California, San Diego, California, USA garsia@math.ucsd.edu D´epartement de math´ematiques et d’informatique Universit´e du Qu´ebec `a Trois-Rivi`eres, Trois-Rivi`eres, Qu´ebec, Canada alain.goupil@uqtr.ca Submitted: Sep 22, 2008; Accepted: Jul 25, 2009; Published: Jul 31, 2009 Mathematics Subject Classifications: 20C30, 20C08, 05E05, 05A18 , 05A15 Dedicated to Anders Bj¨orner on the occasion of his sixtieth birthday. Abstract The numerical calculation of character values a s well as Kronecker coefficients can efficently be carried out by means of character polynomi- als. Yet these po lynomials do not seem to have been given a proper role in present day literature or software. To show their remarkable simplicity we give here an “umbral” version and a recursive combinatorial construction. We also show that these polynomials have a natural counterpart in the standard Hecke algebra H n (q ). Their relation to Kronecker products is brought to the fore, as well as special cases and applications. This paper may also be used as a tutorial for working with character polynomials in the computation of Kronecker coefficients. I. Introduction We recall that the value χ λ α of the irreducible S n character indexed by a partition λ = (λ 1 , . . ., λ k ) at a permutation of S n with cycle structure α = 1 a 1 2 a 2 · · · n a n is g iven by the Frobenius formula χ λ α = ∆(x)p α    x λ 1 +n−1 1 x λ 2 +n−2 2 ···x λ n +n−n n I.1 * Work suppo rted by a grant from NSF. ** Work partially supported by a grant from NSERC. the electronic journal of combinatorics 16(2) (2009), #R19 1 where ∆(x) = ∆(x 1 , . . ., x n ) and p α = p α (x 1 , . . ., x n ) denote respectively the Vander- monde determinant and the power sums symmetric functions. The character polynomial q µ (x 1 , x 2 , . . ., x n ) is the unique polynomial in Q[x 1 , x 2 , . . ., x n ] with t he property that for all partitions µ ⊢ k and λ = (n − k, µ) with n − k ≥ µ 1 we have χ (n−k,µ) 1 a 1 2 a 2 n a n = q µ (a 1 , a 2 , . . ., a n ). I.2 Moreover, wit h an appropriate change of sign and rearrangements of the parts of (n − k, µ) this can be shown to remain true even when n − k < µ 1 . The simplest case of equation I.2 is the well known formula q 1 (x) = x 1 − 1 I.3 which implies that for all n ≥ 2 the value of the character χ (n−1,1) at a permutation σ ∈ S n is simply equal to one less than the number x 1 of fixed points of σ. Character polynomials were implicitly used for the first time in the work of Murnaghan [Mu] and were identified as such later by Specht in [Sp] where a determinantal form for character polynomials and a proof of equation I.2 are given. Treatments of character polynomials vary from the purely existential as in Kerber [Ke], to the very explicit as in Macdonald’ s book ([Ma] ex. I.7.13 and I.7.14). What is quite surprising, as we see in I.3, is how littl e information from the cycle structure may be needed to compute the whole sequence of character values χ (n−k,µ) . We will show here that a slight addition to the computation carried out in [Ma] yields a formula of utmost simplicity which brings to explicit evidence this minimal dependence on the cycle structure. To state our formula let us denote by “↓” the “umbral ” operator that transforms a monomial into a product of lower factorial polynomials. To be precise we set ↓ x a = ↓ x a 1 1 x a 2 2 · · · x a m m = (x 1 ) a 1 (x 2 ) a 2 · · · (x m ) a m I.4 with (x) a = x(x − 1)(x − 2) · · · (x − a + 1). This given we have Proposition I.1 For all µ ⊢ k, the character polynomial q µ depends at most on the first k variables. More precisely q µ (x 1 , x 2 , · · ·, x k ) = ↓  α⊢k χ µ α z α k  i=1 (ix i − 1) a i I.5 where z α = 1 a 1 2 a 2 · · · k a k a 1 !a 2 ! · · ·a k !. In other words, equation I.5 states that we obtain any character polynomial q µ by the following easy sequential steps: the electronic journal of combinatorics 16(2) (2009), #R19 2 a) Expand the Schur function s µ in the power sums basis : s µ =  α⊢k χ µ α z α p α . b) Replace each power sum p i by ix i − 1. c) Expand each product  i (ix i − 1) a i as a sum  g c g  i x g i i . d) Replace each x g i i by (x i ) g i . Let us compute the polynomial q (3) (x 1 , x 2 , x 3 ) by using the preceding steps : a) s 3 = 1 6 (p 1 3 + 3p (21) + 2p (3) ) b) 1 6 (p 1 3 + 3p (21) + 2p (3) ) → 1 6 ((x 1 − 1) 3 + 3(2x 2 − 1) (x 1 − 1) + 2(3x 3 − 1) ) c) 1 6 ((x 1 − 1) 3 + 3(2x 2 − 1) (x 1 − 1) + 2(3x 3 − 1)) = 1 6 x 3 1 − 1 2 x 2 1 + x 1 x 2 − x 2 + x 3 d) q (3) (x 1 , x 2 , x 3 ) = 1 6 (x 1 ) 3 − 1 2 (x 1 ) 2 + x 1 x 2 − x 2 + x 3 Note that when we set x 1 = n in q µ and x i = 0 for all i > 1, we obtain the number f (n−k,µ) of standard t abl eaux of shape (n − k, µ). In view of the classical hook formula, this must reduce to the identity f (n−k,µ) = q µ (x 1 , 0, 0, . . .)    x 1 =n = (x 1 ) k+µ 1 f µ /k!  µ 1 i=1 (x 1 − k + µ ′ i − i + 1)     x 1 =n I.6 ∀ µ ⊢ k & n − k ≥ µ 1 , where µ ′ = (µ ′ 1 , µ ′ 2 , . . .) is the conjugat e partition of µ. An immediate consequence of equation I.5 is a recursive algorithm for the con- struction of the character polynomials that does not directly involve any of the sym- metric group characters. Corollary I.1 For a given partition µ ⊢ k, let ˜µ denote the partitio n obtained by removing the first part from µ. Then q µ (x 1 , x 2 , · · ·, x k ) = ↓  1a 1 +2a 2 +···+ka k =k q ˜µ (a 1 , a 2 , . . ., a k ) 1 a 1 2 a 2 · · · k a k a 1 !a 2 ! · · ·a k ! k  i=1 (ix i − 1) a i with initial setting q ∅ (a 1 , a 2 , . . ., a k ) = 1 . Our next result is a combinatorial formula for the character polynomials which has some kinship with the Murnaghan-Nakayama rule, and is quite suitable for hand calculations. Theorem I.1 For a given µ ⊢ k, let bd(i, µ) be the maximal number of border strips of length i that can successively be removed from the diagram of µ so that a Ferrers diagram remains. Then the terms in q µ that contain the variable x i , 2 ≤ i ≤ k but no variable x r with r > i is given by the recursive rule q µ (x 1 , . . .,x i , 0, . . .) − q µ (x 1 , . . ., x i−1 , 0, . . .) = bd(i,µ)  j=1  x i j   S=(µ 0 ,µ 1 , ,µ j ) (−1) ht(S) q µ j (x 1 , . . ., x i−1 , 0) the electronic journal of combinatorics 16(2) (2009), #R19 3 where the inner sum is over all (j +1)-tuples S of partitions µ r such that each µ r −µ r+1 is a border strip of length i and ht(S) =  j−1 r=0 (height(µ r − µ r+1 ) − 1). The init ial term q µ (x 1 , 0, . . .) is computed via equation I.6. For instance, the polynomial q (3,1,1) (x 1 , x 2 , . . ., x 5 ) is recursi vely constructed as follows. q (3,1,1) (x 1 , . . ., x 5 ) = q (3,1,1) (x 1 ) + (−1) 0  x 2 1  q (1 3 ) (x 1 ) + (−1) 1  x 2 1  q (3) (x 1 ) + (−1) 1 2  x 2 2  q (1) (x 1 ) + (−1) 2  x 5 1  q ∅ (x 1 , . . ., x 4 ) = x 1 (x 1 − 1) · · · (x 1 − 7) (x 1 − 2) (x 1 − 5)(x 1 − 6) f (3,1,1) 5! + x 2  x 1 − 1 3  − x 2  x 1 (x 1 − 1) · · · (x 1 − 5) (x 1 − 2)(x 1 − 3) (x 1 − 4) f (3) 3!  − 2  x 2 2  (x 1 − 1) + x 5 where the last equality follows from equat ion I.6. This pap er is organized as follows. In the first section we introduce o ur notat ion, make some definitions and prove some a ux iliary facts. In the second section we treat the classical S n case, prove our umbral formula for the character polynomials as well as Theorem I .1. In the thi rd section, striv ing to make our writing accessible to a wider audience, we give a brief tutorial on Kronecker products including simple proofs of some basic results of the theory. The experts in symmetric function theory may skip this section. In the fourth section we use the pairing s µ →q µ to define a degree preserving isomorphism that sends t he vector space Λ of symmetric polynomials onto the vector space of polynomials Q[x 1 , x 2 , x 3 , . . .]. We then use this map to derive some well known and some not so well known properties of Kronecker products. The study of this map leads to another family of polynomials that we call “set polynomials ” which enjoy properti es akin to those of character po lynomials and can also be used to compute Kronecker products. In the fifth section we treat the Hecke a lgebra case a nd derive our q-analogs of chara cter polynomials. We present comparative tables of character and q-character po lynomials. In the sixth and last section we explore some consequences of our techniques and give some applications. In particular we obtain an explicit formula yielding a generating function for the occurence of s (n) in Kronecker powers of h r h n−r for every fixed r ≥ 1. This generating function may be viewed as a solutio n to a problem first formulated by Comtet in [Co], namely the enumeration of coverings of a set of cardinality n by sets of cardinality r. The corresponding generating function for r = 2 was first given by Labelle in [La]. A surprisingly simple argument yields the general result and in particular the Label le result. The calculation of character polynomials also yield the electronic journal of combinatorics 16(2) (2009), #R19 4 unexp ected results. For instance, it comes out that the polynomial  k s=0 (−1) k−s n(n − 1) · · ·(n−s+1) enumerates, for n ≥ 2k, the number of permutations σ ∈ S n with longest increasing subsequence σ(1), σ(2), · · · , σ(n − k) = n. A direct proof of this makes an amusing combinatorial exercise. A second unexpected outcome is a formula for t he well known Bell numbers that does not seem to appear in the literature. We terminate the paper with the computation of certain remarkable character polynomials. Acknowledgement. The authors wish to thank Alain Lascoux for his helpful suggestions and remarks on an earlier version of this paper. We also thank the referees for their excellent review. 1. Definitions and basic concepts To begin it will be convenient to write a partition α of n as a weakly decreasing list of parts α = ( α 1 ≥ α 2 ≥ . . . α k ≥ 1) or by giving the list of multiplicities of it s parts : α = 1 a 1 2 a 2 . . . n a n . We will use greek letters λ, µ, α, . . . for the partiti ons and their parts and the corresponding roman l etters ℓ i , m i , a i for their multipli ci ties. The number k of parts of a partition α = (α 1 , α 2 , . . ., α k ) is called the length of α and is denoted ℓ (α). The weight |α| of a partition α i s the sum of its parts and we extend thi s convention to any vector a = (a 1 , . . ., a k ). The expression z α = 1 a 1 2 a 2 · · · n a n a 1 ! · · ·a n ! will be used throughout the text. Thus for a partition α of n we will use the notations α = ( α 1 , α 2 , . . ., α k ) = 1 a 1 2 a 2 . . . n a n , α ⊢ n, |α| = a = 1a 1 + 2a 2 + · · · + na n = n; ℓ(α) = k = |a| = a 1 + a 2 + . . . + a n . 1.1 We will need to merge partitions and for α = 1 a 1 2 a 2 . . . n a n , β = 1 b 1 2 b 2 . . . m b m and n ≥ m we use the operatio n α ∨ β = 1 a 1 +b 1 2 a 2 +b 2 . . . n a n +b n . As customary if µ is a partition, µ ′ will denote the conjugate partition. It will also be convenient to denote by Λ, the space of symmetric functions and by Λ =k the subspace of symmetric functions that are homogeneous of degree k. The number of vari abl es in a symmetric function will always be assumed to be greater or equal to its degree. The Hall scalar product of symmetric functions, with respect to which the Schur functions form an orthonormal system, will be denot ed  ,  . In this paper we shall make extensive use of plethystic nota t ion. The first author has been a proponent of this device since the early 1980’s after Lascoux showed that many identities in Macdonald’s book could acquire a remarkable simplicity in terms of it. This notwithstanding, its use doesn’t seem to have yet achi eved widespread acceptance. This work will give us one more opportunity to show the power of this the electronic journal of combinatorics 16(2) (2009), #R19 5 notational devi ce in the theory of symmetric functions. To this end we recall that the plethystic substitution of a formal power series E(t 1 , t 2 , . . . , t s , . . .) into a symmetric polynomial A, denoted “A[E]” is obtained by setting A[E] = Q A (p 1 , p 2 , . . .)    p k →E(t k 1 ,t k 2 , ) 1.2 where Q A (p 1 , p 2 , . . .) is the polynomial that gives t he expansion of A in terms of the power basis {p α } α . This given, if we let Ω[X] = exp    k≥1 p k [X] k   then setting E = z X n with X n =  n i=1 x i we get Ω[zX n ] = n  i=1 1 (1 − zx i ) =  m≥0 h m [X n ]z m = exp    k≥1 p k [X n ] k z k   1.3 which i s the generating function of t he so-called “homogeneous ” symmetric functions h m . We should note that contrary to intuition the definition in 1.2 yields that p k [−X n ] = p k  − n  i=1 x i  = − n  i=1 x k i = −p k [X n ] This paradox , in spite of being a hindrance, is a n asset. In fact, we need only distinguish the “plethystic” mi nus sign from the customary minus sign. This is easily achieved. When we want to replace a variable x i or a formal power series E by its negative in the customary sense we simply prepend a minus sig n to it as in “ − x i ” or “ − E” a nd use the regular minus sign for plethystic subtraction. Note that with this convention, and X =  i x i we get p k [− − X] = (−1) k−1 p k [X] = ωp k [X] 1.4 where “ω” is the fundamental involution on Λ that sends the homogeneous basis onto the elementary basis. In particular from 1.4 we derive that for any symmetric funtion A[X] we have, when the alphabet X has a sufficient number of variables, ω A[X] = A[− − X]. Moreover, when A is homogeneous of degree k, we see that this is equivalent to A[−X] = (−1) k ω A[X]. 1.5 the electronic journal of combinatorics 16(2) (2009), #R19 6 In the same vein we show that Proposition 1.1 For any partition λ and any two al phabets X =  i x i and Y =  j y j we have the Schur function identity s λ [X − Y ] =  µ⊆λ s λ/µ [X](−1) |µ| s µ ′ [Y ]. 1.6 In particular for any m ≥ 1 we get that h m [X − Y ] = m  k=0 h m−k [X](−1) k e k [Y ] 1.7 and for Y = 1 this specializes to h m [X − 1] = h m [X] − h m−1 [X] 1.8 Proof. Since 1.7 fol lows from 1.6 by setting λ = (m) we need only show 1.6. On the other hand 1 .6 is an immediate consequence of 1.5 and of the addition formula ([Ma], ch. I, 5.9) s λ [X + Y ] =  µ⊆λ s λ/µ [X]s µ [Y ]. 1.9 upon replacement of Y by −Y . In fact, from 1.5 it follows that s µ [−Y ] = (−1) |µ| s µ ′ [Y ]. 1.10 Alternatively we can prove 1.7 without using the addition formula. Indeed we have Ω[t(X − Y )] = ex p   k≥1 t k p k [X − Y ] k  = exp   k≥1 t k p k [X] k  exp   k≥1 −t k p k [Y ] k  =   a≥0 t a h a [X]   b≥0 (−t) b e b [Y ]  =  n≥0 t n n  k=0 h k [X](−1) n−k e n−k [Y ] and since 1.3 gives Ω[t(X − Y )] =  n≥0 t n h n [X − Y ], we o bta in 1.7 by taking the coefficient of t n . More generally, for any formal power series E we define, the “transl a tion by E ” oper- ator, “T E ”, on symmetric functions A[X] by T E A[X] = A[X + E] the electronic journal of combinatorics 16(2) (2009), #R19 7 Now for any symmetric function A it is customary to denote by A ⊥ the adjoint of the operator multiplicati o n by A, with respect to the Hall scalar product. It follows from the definition of skew Schur functions s λ/µ that we may write s λ/µ = s ⊥ µ s λ . Thus 1.6 and 1.9 may be written in the form T −Y s λ [X] =   µ (−1) |µ| s µ ′ [Y ] s ⊥ µ  s λ [X] and T Y s λ [X] =   µ s µ [Y ] s ⊥ µ  s λ [X] and since Schur functions form a basis, it follows that we may write T −Y =  µ (−1) |µ| s µ ′ [Y ]s ⊥ µ and T Y =  µ s µ [Y ]s ⊥ µ Since Y can be replaced by any formal series we see that these two identities are specia l cases of the expansion T E =  µ s µ [E]s ⊥ µ The particular instances that will play a role here are obtained by setting E = ±1. These simply reduce to T −1 =  k≥0 (−1) k e ⊥ k and T 1 =  k≥0 h ⊥ k 1.11 An interesting generalization of 1.8 is the following Proposition 1.2 Fo r any partition µ = (µ 1 , µ 2 . . . , µ k ) we have s µ [X − 1] = det       1 1 1 · · · 1 h µ 1 −1 h µ 1 h µ 1 +1 · · · h µ 1 +k−1 h µ 2 −2 h µ 2 −1 h µ 2 · · · h µ 2 +k−2 . . . . . . . . . . . . . . . h µ k −k+1 h µ k −k+2 h µ k −k+3 · · · h µ k       1.12 with the usual convention that h m = 0 when m < 0. Proof. It is sufficient to illustrate the argument in a special case. So let µ = (4 , 3, 1). In this case the Jacobi-Trudi identity gives s µ [X] = det   h 4 h 5 h 6 h 2 h 3 h 4 0 1 h 1   . the electronic journal of combinatorics 16(2) (2009), #R19 8 making the substitution X → X − 1 and using 1.8, we get s µ [X − 1] = det   h 4 − h 3 h 5 − h 4 h 6 − h 5 h 2 − h 1 h 3 − h 2 h 4 − h 3 0 1 h 1 − 1   = det    1 1 1 1 h 3 h 4 h 5 h 6 h 1 h 2 h 3 h 4 0 0 1 h 1    . since the first determinant is obtained from the second by subtracting from each column the preceeding column giving det    1 0 0 0 h 3 h 4 − h 3 h 5 − h 4 h 6 − h 5 h 1 h 2 − h 1 h 3 − h 2 h 4 − h 3 0 0 1 h 1 − 1    The general case of equation 1.12 can clearly be established in a similar manner. 2. Proofs of the umbral and recursive formulas After this foray into plethystic magic we are ready to play with character poly- nomials. Our point of departure, as in Macdonald ( [Ma]), is the Frobenius formula, equation I.1, in the vari ables x 0 , x 1 , . . ., x n which for λ = (n − k, µ) with µ ⊢ k may be written in the form χ (n−k,µ) α =  0≤i<j≤n (x i − x j )p α [x 0 + · · · + x n ]    x n−k+n 0 x µ 1 +n−1 1 ···x µ n n 2.1 where for conveni ence we have set µ r = 0 for al l r > ℓ(µ). As in [Ma] (ch. 1 ex 14), we note that the homogeneity in x 0 , x 1 , . . ., x n of the polynomial in 2.1 allows us to set x 0 = 1 and reduce this identity to χ (n−k,µ) α = n  i=1 (1 − x i )  1≤i<j≤n (x i − x j )p α [1 + X n ]    x µ 1 +n−1 1 ···x µ n n = n  i=1 (1 − x i )T 1 p α    s µ =  n  i=1 (1 − x i )T 1 p α , s µ  2.2 where “  ,  ” denotes the Hall scalar product. Since  n i=1 (1 − x i ) =  n r=0 (−1) r e r [X n ] we may rewrite 2.2 as χ (n−k,µ) α =  T 1 p α , n  r=0 (−1) r e ⊥ r s µ  . the electronic journal of combinatorics 16(2) (2009), #R19 9 Using the first identity i n 1.11, this in turn can be rewritten as χ (n−k,µ) α =  T 1 p α , T −1 s µ  . 2.3 Up to this point, albeit with some slight difference of notation, we have followed Mac- donald almost verbatim. To obtain our umbral formula we only need to diverge slightly from Macdonald’s path. To begin we use the expansion T −1 s µ =  β⊢k β=1 b 1 2 b 2 ···k b k χ µ β z β k  i=1 (p i − 1) b i and the second identity in 1.11 to rewrite 2.3 as χ (n−k,µ) α =  β⊢k β=1 b 1 2 b 2 ···k b k χ µ β z β  p α , n  m=0 h m k  i=1 (p i − 1) b i  . 2.4 Since the operator adjoint to multiplication by p i is simply i∂ p i , we derive that  p α , n  m=0 h m k  i=1 (p i − 1) b i  =  k  i=1 (i∂p i − 1) b i p α , n  m=0 h m  . 2.5 Now note that for any integral vector t = (t 1 , t 2 , . . ., t n ) we have  n  i=1 ∂ t i p i p α , n  m=0 h m  = n  i=1 (a i ) t i  n  i=1 p a i −t i i , n  m=0 h m  and the identity  p α , h m  =  1 if α ⊢ m, 0 otherwise. gives  n  i=1 ∂ t i p i p α , n  m=0 h m  = n  i=1 (a i ) t i = ↓ n  i=1 x t i i    x i =a i , Expanding the first product in the right hand side of 2. 5 we obtain  p α , n  m=0 h m k  i=1 (p i − 1) b i  = ↓ k  i=1 (ix i − 1) b i    x i =a i . the electronic journal of combinatorics 16(2) (2009), #R19 10 [...]... yields the expansion sµ ∗ sν = cλ,µ,ν sλ 3.6 λ⊢n with χλ χµ χν /zα α α α cλ,µ,ν = sµ ∗ sν , sλ = 3.7 α⊢n where the last equality is obtained by combining equation 3.5 with 3.2 and the relation pα , pβ = zα 0 the electronic journal of combinatorics 16(2) (2009), #R19 if α = β, if α = β 3.8 14 The cλ,µ,ν go by the name of Kronecker coefficients” and they may be interpreted as the multiplicity of the irreducible... relation in equation 3.6 enables us to attack the Kronecker coefficient problem by symmetric function methods In this writing we will focus on the identities that reduce the computation of the Kronecker products of symmetric functions to ordinary products and thereby ultimately express the cλ,µ,ν in terms of the gλ,µ,ν To this end it is convenient to extend the Kronecker product to all of Λ by setting P... 4.8, pops out only in the second step of the algorithm In summary, to assure that equation 4.12 remains true, the straightening of terms sn−|λ|,λ causes terms with partition indexing to disappear from the right-hand side, and this is precisely what causes the delay in the stabilization of the Kronecker coefficients 5 The Hecke algebra case In [Ra] Arun Ram obtains a q-analog of the Frobenius formula... 3.1 and 3.4 when α and β are partitions of the same number and from equation 3.9 when they are not This given, 2) follows by linearity from equation 3.2 Finally, 3) follows from equation 3.9 together with 1) and the well known expansion hn = pα /zα 3.11 α⊢n the electronic journal of combinatorics 16(2) (2009), #R19 15 The following basic identity of Littlewood ([Li]) provides an algorithm for the. .. ] i=1 hi →q i−1 (q−1) and equation 5.24 follows from equation 5.18 Table 5.1, where ℓ stands for ℓ(α), should (n−k,µ) dispell any doubts that the polynomials yielding the q -character values χα (q) are natural q-analogs of the character polynomials 6 Set polynomials and applications We begin by noting that, using equation 2.8 or I.2, the identity 2.3 may be rewritten in any of the following two equivalent... polynomials” the polynomials v(f ) defined by equation 6.7 It is well known that the number of occurrences of the identity representation in the action of a group G on a family F of objects is equal to the number of orbits of this action If G acts transitively then its action on F is equivalent to its action on the left cosets of the stabilizer of any one of the elements of F It follows then that if G = Sn , the. .. group of permutations of Ωn = {1, 2, , n} and if F is the family of r-subsets of Ωn then the Frobenius characteristic of the action of Sn on F is the symmetric function hr hn−r Denoting by Πn−r,r the character of this action evaluated on the permutations α of cycle type α we must have α Πn−r,r pα α z hn−r hr = 6.10 α⊢n = hn−r hr , pα pα zα 6.11 α⊢n the electronic journal of combinatorics 16(2)... and then equation I.1 followed from the bideterminantal formula for Schur functions The Kronecker product in Λ=n is defined by setting for P, Q ∈ Λ=n P ∗ Q = Fn F−1 (P ) × F−1 (Q) n n 3.4 where the symbol ‘×” here represents the pointwise product in C(Sn ) In particular from equation 3.2 it follows that χµ χν pα /zα α α sµ ∗ sν = (for all µ, ν ⊢ n) 3.5 α⊢n The orthonormality of the Schur functions then... all the other character values are obtained as linear combinations of the polynomials χλ (q) Remarkably, Ram is also α λ able to show that the polynomials χα (q) may be computed manually by an algorithm that is essentially a q-analog of the Murnaghan-Nakayama rule The methods we developped in the Sn case can also be extended to Hn (q) thereby obtaining an explicit, albeit not as simple, q-analog of the. .. ) xi =···=xk =0 and equation 2.12 becomes k qµ (x1 , , xi ) − qµ (x1 , · · · , xi−1 ) qµ (x1 , x2 , , xk ) = qµ (x1 ) + i=2 the electronic journal of combinatorics 16(2) (2009), #R19 13 which is obviously true 3 Basics on the Kronecker product Let Cα denote the formal sum of the permutations of Sn with cycle type α and let C(Sn ) denote the center of the group algebra of Sn The map Fn : C(Sn . Character Polynomials, their q-analogs and the Kronecker product A. M. Garsia* and A. Goupil ** Department of Mathematics University of California, San. x n ) and p α = p α (x 1 , . . ., x n ) denote respectively the Vander- monde determinant and the power sums symmetric functions. The character polynomial q µ (x 1 , x 2 , . . ., x n ) is the unique. true. 3. Basics on the Kronecker product Let C α denote the formal sum of the permutations of S n with cycle type α and let C(S n ) denote the center of the group algebra of S n . The map F n : C(S n )