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A Plethysm Formula on the Characteristic Map of Induced Linear Characters from Un(Fq) to Gln(Fq)

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A Plethysm Formula on the Characteristic Map of Induced Linear Characters from Un(Fq) to Gln(Fq) A Plethysm formula on the characteristic map of induced linear characters from Un(Fq) to GLn(Fq) Zhi Chen Department of Mathematics and Statistics York University Toronto, ON, Canada [email protected] Submitted: May 19, 2012; Accepted: Aug 2, 2013; Published: Aug 9, 2013 Mathematics Subject Classifications: 05E10, 20C33 Abstract This paper gives a plethysm formula on the characteristic map of the induced linear characters from the unipotent upper-triangular matrices Un(Fq) to GLn(Fq), the general linear group over finite field Fq. The result turns out to be a multiple of a twisted version of the Hall-Littlewood symmetric functions P~n[Y ; q]. A recurrence relation is also given which makes it easy to carry out the computation. Keywords: representation theory; induced characters; symmetric functions; su- percharacter theory 1 Introduction Let Fq be a fixed finite field and GLn(Fq) the finite general linear group over Fq. The representation theory of GLn(Fq) over C was presented by J.A.Green [5]. He also con- structed the characteristic map which builds a connection between the character spaces of GLn(Fq) for n > 0 and the Cartesian product over infinitely indexed sets of rings of symmetric functions. In character theory, the study of induced linear characters from subgroups is very useful in order to understand the character ring of the larger group. In this paper, we consider certain induced linear characters from the group of unipo- tent upper-triangular matrices Un(Fq) to GLn(Fq). The representations of these induced linear characters are known as Gelfand-Graev modules, which play an important role in the representation theory of finite groups of Lie type ([4], [12]). The formula for the characteristic map of the induced linear characters is given by Thiem [9]. We then apply the electronic journal of combinatorics 20(3) (2013), #P12 1 a homomorphism ρ (see Definition 5) on the image of the characteristic map. We refer to the result as a plethysm formula because it involves a composition of symmetric functions using the plethysm operation. There are two advantages in doing so: to get a simpler formula and to express the result as a multiple of a twisted version of the Hall-Littlewood ~ symmetric functions Pn[Y ; q]. We hope this method could contribute to the study of the irreducible decomposition of the induced characters from Un(Fq) to GLn(Fq). In section 2 we give some background knowledge on symmetric functions and repre- sentation theory of GLn(Fq) and Un(Fq). Since the character theory of Un(Fq) is known as a wild problem, supercharacter theory is built up as an approximation of the ordinary character theory. The linear characters of Un(Fq) that we are considering are part of the category of supercharcters of Un(Fq). We introduce further questions about the induc- tion of all supercharacters in Section 4. In Section 3 we give our main result about the plethysm formula. A natural recurrence relation is obtained so that we can carry out the computation of the homomorphism ρ on the characteristic map of the induced linear char- acters more easily. We also give a relation between the characteristic map of the induced characters from Un(Fq) to GLn(Fq), and the homomorphism ρ on those characteristics. This is depicted in the following diagram ' Id ⊗'2ΘΛC(Y ) ⊗f2ΦΛC(Xf ) ρ ΠjΛ (X ) C f=x−1 ΛC(Y ) τ◦! / ΛC(Xx−1) where the notation is explained in Theorem 17. From the above commutative diagram we show that our simplified plethysm formula does not lose any information on the char- acteristic map of the induced characters from Un(Fq) to GLn(Fq). 2 Background 2.1 Symmetric functions The notation in this paper follows closely the book of Macdonald [7] and Thiem [10]. Definition 1. A partition λ of n 2 N, is a sequence λ = (λ1; λ2; : : : ; λl) of positive integers in weakly decreasing order: λ1 > λ2 > ··· > λl, such that λ1 + λ2 + ··· + λl = n. We denote this by λ ` n. Here, each λi (1 6 i 6 l) is called a part of λ. We say the length of the partition λ is l = l(λ), which is the number of parts of λ. We use jλj to denote the sum of all parts λ1 + λ2 + ··· + λl, and we call jλj the size of the partition. Sometimes we also use the notation: λ = (1m1 ; 2m2 ; : : : ; nmn ;:::); where each mi means there are mi parts in λ equal to i. Let ΛC(Y ) denote the ring of symmetric functions with complex coefficients in the variables Y = fy1; y2;:::g. We denote the complete symmetric functions, elementary the electronic journal of combinatorics 20(3) (2013), #P12 2 symmetric functions, monomial symmetric functions, power-sum symmetric functions, and Schur symmetric functions by hλ[Y ], eλ[Y ], mλ[Y ], pλ[Y ], and sλ[Y ] respectively. Following other references (e.g. Garsia and Tesler [3]), we use the notation Ω for the basic symmetric function kernel. Then the generating function for hn[Y ] is X n Y −1 Ω[tY ] = hn[Y ]t = (1 − yjt) : n>0 j>1 It is also well known that Ω[X]Ω[Y ] = Ω[X + Y ]; and Ω[X]=Ω[Y] = Ω[X − Y]: Let X = fx1; x2;:::g be another set of finite or infinite variables. We have the following identity: Y −1 X Ω[XY ] := (1 − xiyj) = mλ[X]hλ[Y ] (2.1) i;j λ summed over all partitions λ. There is a scalar product defined on ΛC(Y ), which makes (mλ) and (hλ) dual to each other: hhλ; mµi = δλµ for all partitions λ, µ, where δλµ is the Kronecker delta. We use Pλ[Y ; t] to denote the Hall-Littlewood symmetric functions, as defined in [7] (see page 209). If we define qr = qr[Y ; t] = (1 − t)P(r)[Y ; t] for r > 1; q0 = q0[Y ; t] = 1; then the generating function for qr[Y ; t] is X r Y 1 − yitu Q(u) = qr[Y ; t]u = = Ω[(1 − t)uY ] (2.2) 1 − yiu . r>0 i For each partition λ, let n(λ) = P (i − 1)λ . Define i>1 i ~ −n(λ) −1 Pλ[Y ; q] = q Pλ[Y ; q ]; ~ and we call Pλ[Y ; q] the twisted Hall-Littlewood symmetric functions. From [7], it is well known that the plethysm can be defined by b b ha[pb[Y ]] = ha[y1; y2;:::]; (2.3) which is the coefficient of tab in Q (1 − ybtb)−1: j>1 j the electronic journal of combinatorics 20(3) (2013), #P12 3 2.2 Representation theory of GLn(Fq) The representation theory of the finite general linear group Gn = GLn(Fq) over C can be found in J.A.Green [5], Macdonald [7] and Thiem [9]. Here we give a short description of the characteristic map constructed by J.A.Green. ¯ Let Fq denote the algebraic closure of the finite field Fq. The multiplicative group of ¯ ¯× ¯ Fq is denoted by Fq . The Frobenius automorphism of Fq over Fq is given by q ¯ F : x ! x ; where x 2 Fq: ¯× n ¯× For each n > 1, we use Fq;n to denote the fixed points of F in Fq . ¯∗ ¯× × Let Fq = fφ : Fq ! C g be the group of complex-valued multiplicative characters of ¯× ¯∗ Fq . The Frobenius automorphism on Fq is q ¯∗ F : ξ ! ξ ; where ξ 2 Fq: ¯∗ n For each n > 1, let Fq;n be the group of elements fixed by F . We also define a pairing of ¯∗ ¯× Fq with Fq by hξ; xin = ξ(x) ¯∗ ¯× for ξ 2 Fq;n and x 2 Fq;n. We then define ¯× ¯∗ Φ = fF -orbits of Fq g and Θ = fF -orbits of Fqg: ¯× Since each F -orbit of Fq is in one-to-one correspondence with an irreducible polynomial over Fq, we can also use f to denote each F -orbit in Φ. A partition-valued function µ on Φ is a function which maps each f 2 Φ to a partition µ(f). The size of µ is X kµk = d(f)jµ(f)j; f2Φ where d(f) is equal to the degree of f 2 Φ. Let P denote the set of all partitions and Φ [ Φ Φ P = Pn ; where Pn = fµ :Φ ! P ; kµk = ng: n>0 µ Φ We use K to denote the conjugacy classes in Gn parameterized by µ 2 Pn [7]. The µ characteristic function of the conjugacy class K is denoted by πµ. Similarly, for each partition-valued function λ :Θ ! P, the size of λ is X kλk = d(')jλ(')j; '2Θ where d(') is equal to the number of elements in '. Let Θ [ Θ Θ P = Pn ; where Pn = fλ :Θ ! P ; kλk = ng: n>0 the electronic journal of combinatorics 20(3) (2013), #P12 4 λ Θ We use Gn to denote the irreducible Gn-modules indexed by λ 2 Pn [7]. The character λ λ of the irreducible Gn-module Gn is denoted by χ . For every f 2 Φ, let Xf := fX1;f ;X2;f ;:::g be a set of infinitely many variables. Each Xi;f has degree d(f). Let ~ ~ d(f) −d(f)n(η) −d(f) Pη[f] = Pη[Xf ; q ] = q Pη[Xf ; q ]; ~ d(f) where Pη[Xf ; q ] is the twisted Hall-Littlewood symmetric function. Define ~ Y ~ Pµ = Pµ(f)[f]: f2Φ ' ' ' For every ' 2 Θ, let Y := fY1 ;Y2 ;:::g be a set of infinitely many variables. Each ' Yi has degree d('). Define Y ' Sλ = sλ(')[Y ]; '2Θ ' where sλ(')[Y ] is the Schur symmetric function.
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