International Journal of Algebra, Vol. 3, 2009, no. 1, 1 - 19 Wreath Product Symmetric Functions Frank Ingram Mathematics Department, Winston-Salem State University Winston-Salem, NC 27110, USA [email protected] Naihuan Jing1 School of Sciences, South China University of Technology Guangzhou 510640, China Department of Mathematics, North Carolina State University Raleigh, NC 27695-8205, USA [email protected] Ernie Stitzinger Department of Mathematics, North Carolina State University Raleigh, NC 27695-8205, USA [email protected] Abstract We systematically study wreath product Schur functions and give a combinatorial construction using colored partitions and tableaux. The Pieri rule and the Littlewood-Richardson rule are studied. We also discuss the connection with representations of generalized symmetric groups. Mathematics Subject Classi¯cation: 05E10, 20E22, 20C30 Keywords: Symmetric funcions, wreath products, Schur functions 1 Introduction Since Jacobi and Frobenius, symmetric functions have played a fundamen- tal role in representations theory and number theory. The character formula 1Corresponding author 2 F. Ingram, N. Jing and E. Stitzinger of the symmetric group in terms of Schur functions and power sums is un- doubtedly one of the most beautiful chapters in group theory and algebraic combinatorics. Schur developed Frobenius theory further using his rational ho- mogeneous functions, and again symmetric functions were the most important techniques in his work on generalizations of Frobenius character theory. In Macdonald's book [4], one can see other interesting examples of ap- plications of symmetric functions into group theory. Among them, Specht's generalization [9] of Schur functions deserves special mention. Let G be any n ¯nite group, the semidirect product G n Sn denoted by G » Sn is called the wreath product of the ¯nite group G and the symmetric group Sn. The special case 1 » Sn is exactly the symmetric group Sn, while Z2 » Sn is the hyperoctahedral group. Recently the wreath products are used in the gener- alized McKay correspondence [1] to realize the a±ne Lie algebras of simply laced types. The wreath products have also been studied in various interest- ing contexts [6, 2, 5]. However the basic formulation of the characters remains complicated and formidable. It is well-known that Schur functions can also be de¯ned in terms of Young tableaux in Stanley's work [10]. One purpose of this paper is to give a simpler and more elementary description of wreath product symmetric functions using colored tableaux. We achieve this by ¯rst considering tensor product of Schur functions and then pass the results to the case of wreath product using linear transformations. In our simpli¯ed approach one can see the structures of wreath product symmetric functions easily. In the same spirit we also obtain the description of the Littlewood-Richardson rule for the wreath product Schur functions. We deviate slightly from Macdonald's inner product by requiring that the power sum associated with irreducible characters of G to be invariant under the complex conjugation, which will facilitate us to pass the results from tensor product to wreath product easily. The paper is organized into ¯ve parts. After introduction (section one) we give a preliminary review of the main results concerning Schur functions to set the notations and for later usage in Section two. In Section three we studied tensor products of symmetric functions and formulate several bases modeled on one variable case. The notion of colored partitions and tableaux is the fundamental combinatorial objects for us to formulate results in tensor products. In Section four wreath product Schur functions are introduced by two sets of power sum symmetric functions indexed by colored partitions, which correspond to conjugacy classes and irreducible characters of G » Sn. We show how to pass from tensor products to wreath products by carefully treating the inner product. In section ¯ve we give generalized Littlewood-Richardson rule and discuss their special cases, and ¯nally we discuss the relationship between symmetric functions and characters of wreath products of symmetric groups. Wreath product symmetric functions 3 2 Symmetric functions In this section we review some basic materials on symmetric functions that will be used in our later discussion. Almost all statements are standard and can be found in [4] or in a slightly di®erent notation in [10]. A partition ¸ of an non-negative integer n, denoted as ¸ ` n, is an integral decomposition of n arranged in decreasing order: n = ¸1 + ::: + ¸l, ¸1 ¸ ::: ¸ ¸l ¸ 0. If the last part ¸l 6= 0, the partition has length l(¸) = l, and we also denote by j¸j = n. One sometimes uses the notation ¸ = (1m1 2m2 ¢ ¢ ¢ ) if there are mi parts equal to i in ¸. The Ferrer diagram associated with the partition ¸ ` n with l(¸) = l consists of l layers of n boxes aligned to the left from the top to the bottom. The partition corresponding to the flip of the Ferrer diagram of ¸ along the northwest-to-southeast axis is called the conjugate partition ¸0. For example the Ferrer diagram (4322) and its conjugate diagram are shown in Figure 1. Figure 1. ¸ = (4; 3; 2; 2) and ¸0 = (4; 4; 2; 1) There are two partial orders in the set of partitions. The ¯rst one is the dominance order ¸. Let ¸ and ¹ be two partitions, ¸ ¸ ¹ means that ¸1 ¸ ¹1, ¸1 + ¸2 ¸ ¹1 + ¹2, etc. The second one is the containment order ⊃. If all parts of ¸ are larger than corresponding parts of ¹, we say that ¸ contains ¹ and denote by ¸ ⊃ ¹. Let ¤Z be the ring of symmetric functions in the xi(i 2 N) over Z. For each non-negative integer n, the elementary symmetric function en is de¯ned by X en = xi1 ¢ ¢ ¢ xin ; i1<¢¢¢<in where the sum is taken formally, or it is in the sense of the inverse direct limit determined by the relations: em(x1; : : : ; xn; 0;:::; 0) = en(x1; : : : ; xn) whenever m ¸ n. For each partition ¸ with l(¸) · n the monomial symmetric function m¸ is de¯ned by m (x ; : : : ; x ) = x¸1 ¢ ¢ ¢ x¸n + distinct permutations, where we ¸ 1 n 1 ¸n take ¸l+1 = ¢ ¢ ¢ = ¸n = 0 if necessary. 4 F. Ingram, N. Jing and E. Stitzinger For each partition ¸ we de¯ne e¸ = e¸1 ¢ ¢ ¢ e¸l ; m¸ = m¸1 ¢ ¢ ¢ m¸l : P The homogeneous symmetric function hn is de¯ned by hn = j¸j=n m¸. For any partition ¸ we de¯ne h¸ = h¸1 ¢ ¢ ¢ h¸l As a graded vector space we have M M M ¤Z = Ze¸ = Zm¸ = Zh¸; (1) ¸ ¸ ¸ where the sums are over all partitions. Moreover ¤Z = Z[e1; e2;:::] = Z[h1; h2;:::] as polynomials rings, and feng and fhng are algebraically independent over Z. Subsequently the sets fe¸g, fm¸g and fh¸g are all Z-linear bases of ¤Z. For each non-negative integer n, the power sum symmetric function pn = P n i xi is an element in ¤Q, and we de¯ne for each partition ¸ p¸ = p¸1 ¢ ¢ ¢ p¸l : (2) It is well-known that the fp¸g forms a Q-linear basis in ¤Q, and subsequently ¤Q = Q[p1; p2;:::]. Moreover the pn are algebraically independent over Q. The space ¤Z has a Z-valued bilinear form < ; > de¯ned by < h¸; m¹ >= ±¸¹: (3) Equivalently the symmetric bilinear form can also be de¯ned by < p¸; p¹ >= ±¸¹z¹; (4) Q mi where z¸ = i i mi!. The distinguished orthonormal basis{Schur functions are given by the following triangular relations [4]: X s¸ = m¸ + c¸¹m¹; for some c¸¹ 2 Z ¹<¸ < s¸; s¹ >= ±¸¹ Let ¸ and ¹ be partitions such that ¹ ½ ¸. The skew diagram ¸=¹ is the set-theoretic di®erence ¸ ¡ ¹. A maximal connected component of ¸=¹ is called a connected component. Each connected component itself is a skew diagram. For example the skew Ferrer diagram ¸=¹ =(6,4,2,2)/(4,2) has three components and is shown by Figure 2. A horizontal strip is a skew Ferrer diagram ¸=¹ with no two boxes in the 0 0 same column, i.e., ¸i ¡ ¹i · 1. A vertical strip is skew Ferrer diagram ¸=¹ with no two boxes in the same row, i.e., i.e., ¸i ¡ ¹i · 1 Wreath product symmetric functions 5 Figure 2. Skew diagram (6422)=(42) The multiplication of two Schur functions is again a symmetric function and thus can be expressed as a linear combination of Schur functions: X º º s¸s¹ = c¸¹sº; c¸¹ 2 Z+: (5) º º The structure constants c¸¹ are called the Littlewood-Richardson coe±cients. º It is known that c¸¹ 6= 0 unless jºj = j¸j + j¹j and º ⊃ ¸; ¹. The skew Schur function s¸=¹ 2 ¤Z is then de¯ned by X ¸ s¸=¹ = c¹ºsº: (6) º It is clear that deg(s¸=¹) = j¸j ¡ j¹j and s¸=¹ = 0 unless ¹ ½ ¸, i.e. s¸=¹ is de¯ned for the skew Ferrer diagram ¸=¹. Let x = (x1; x2;:::) and y = (y1; y2;:::) be two sets of variables. Proposition 2.1 [4] (a)Q If the connected components of the skew diagram ¸=¹ are θi, then s¸=¹(x) = i sθi (x). In particular, s¸=¹(x) = 0 if ¹ * ¸. (b) The skew Schur symmetric function s¸(x; y) satis¯es X s¸=¹(x; y) = s¸=º(x)sº=¹(y); (7) º where the sum runs through all the partition º such that ¸ ⊃ º ⊃ ¹. (1) (n) (c) In general, the skew Schur function s¸=¹(x ; : : : ; x ) can be written as X Yn (1) (n) (i) s¸=¹(x ; : : : ; x ) = sº(i)=º(i¡1) (x ); (8) (º) i=1 where the sum runs through all sequences of partitions (º) = (º(n); : : : ; º(0)) such that ¹ = º(0) ½ º(1) ½ ::: ½ º(n) = ¸. If there is only one variable, the Schur function can be easily computed. Proposition 2.2 [4] (a) We have s¸(x) = 0 when l(¸) > 1.