Representation theory of the α-determinant and zonal spherical functions Kazufumi KIMOTO December 14, 2007 Abstract We prove that the multiplicity of each irreducible component in the U(gln)-cyclic module generated by α l the l-th power det( )(X) of the α-determinant is given by the rank of a matrix whose entries are given by a n variation of the spherical Fourier transformation for (Snl, Sl ). Further, we calculate the matrix explicitly when n = 2. This gives not only another proof of the result by Kimoto-Matsumoto-Wakayama (2007) but also a new aspect of the representation theory of the α-determinants. Keywords: Alpha-determinant, cyclic modules, irreducible decomposition, Gelfand pair, zonal spherical functions, Jacobi polynomials. 2000 Mathematical Subject Classification: 22E47, 43A90. 1 Introduction Let n be a positive integer. We denote by Sn the symmetric group of degree n. For a permutation σ ∈ Sn, we define def ν(σ) = (i − 1)mi(σ) (σ ∈ Sn), Xi≥1 where mi(σ) is the number of i-cycles in the disjoint cycle decomposition of σ. We notice that ν(·) is a class ν(σ) function on Sn. It is easy to see that (−1) = sgn σ is the signature of a permutation σ. (α) Let α be a complex number and A = (aij )1≤i,j≤n an n by n matrix. The α-determinant det (A) of A is defined by (α) def ν(σ) arXiv:0712.2293v1 [math.RT] 14 Dec 2007 det (A) = α aσ(1)1aσ(2)2 ...aσ(n)n. (1.1) S σX∈ n We readily see that the α-determinant det(α)(A) coincides with the determinant det(A) (resp. permanent per(A)) of A when α = −1 (resp. α = 1). Hence we regard the α-determinant as a common generalization of the determinant and permanent. The α-determinant is first introduced by Vere-Jones [11]. He proved the identity ∞ ai1i1 ... ai1ik 1 det(I − αA)−1/α = det(α) . .. (1.2) k! . k=0 1≤i1,...,ik≤n a ... a X X ik i1 ikik for an n by n matrix A = (aij )1≤i,j≤n such that the absolute value of any eigenvalue of A is less than 1. Here I denotes the identity matrix of suitable size. His intention of the study of the α-determinant is an application to probability theory. Actually, the identity (1.2) supplies a unified treatment of the multivariate binomial and negative binomial distributions. Further, Shirai and Takahashi [10] proved a Fredholm determinant version of (1.2) for a trace class integral operator and use it to define a certain one-parameter family of point processes. We note that a pfaffian analogue of the Vere-Jones identity (1.2) has been also established and is applied to 1 2 K. Kimoto i−1 probability theory by Matsumoto [7]. It is also worth noting that (1.2) is obtained by specializing pi(x)= α and regarding y1,...,yn as eigenvalues of A in the Cauchy identity 1 1 = pλ(x)pλ(y), (1.3) 1 − xiyj zλ i,jY≥1 Xλ where λ in the right-hand side runs over the set of all partitions, zλ denotes the cardinality of the centralizer of a permutation whose cycle type is λ, and pλ denotes the power-sum symmetric function corresponding to λ (see [6] for detailed information on symmetric functions). In fact, under the specialization, the left-hand side of (1.3) becomes det(1 − αt)−1/α and the right-hand side represents its expansion in terms of α-determinants (see also [3]). In this article, we focus our attention on the representation-theoretic aspect of the α-determinant. Let C U(gln) be the universal enveloping algebra of the general linear Lie algebra gln = gln( ), and P(Matn) be the 2 polynomial algebra in the n variable xij (1 ≤ i, j ≤ n). We put X = (xij )1≤i,j≤n and write an element in P(Matn) as f(X) in short. The algebra P(Matn) becomes a left U(gln)-module via n ∂f(X) E · f(X)= x ij is ∂x s=1 js X for f(X) ∈ P(Matn) where {Eij }1≤i,j≤n is the standard basis of gln. Now we regard the α-determinant (α) det (X) of X as an element in P(Matn) and consider the cyclic submodule V def (α) l n,l(α) = U(gln) · det (X) of P(Matn). Since V ∼ (1n) V ∼ (n) n,1(−1) = U(gln) · det(X) = Mn , n,1(1) = U(gln) · det(X) = Mn , (1.4) the module Vn,1(α) is regarded as an interpolation of these two irreducible representations. Here we denote λ by Mn the irreducible U(gln)-module whose highest weight is λ. We notice that we can identify the dominant integral weights with partitions as far as we consider the polynomial representations of U(gln). Our main concern is to solve the V Problem 1.1. Describe the irreducible decomposition of the U(gln)-module n,l(α) explicitly. In [4], the following general result on Vn,l(α) is proved. F λ Theorem 1.2. For each λ ⊢ nl such that ℓ(λ) ≤ n, there exists a certain square matrix n,l(α) of size Kλ(ln) whose entries are polynomials in α such that F λ V ∼ λ ⊕ rk n,l(α) n,l(α) = (Mn) . λ⊢nl ℓ(Mλ)≤n Here Kλµ denotes the Kostka number and ℓ(λ) is the length of λ. F λ V We call this matrix n,l(α) the transition matrix for λ in n,l(α). We notice that the transition matrix is F λ determined up to conjugacy. Thus, Problem 1.1 is reduced to the determination of the matrices n,l(α) relative F λ to a certain (nicely chosen) basis. Up to the present, we have obtained an explicit form of n,l(α) in only several particular cases. Example 1.3. When l = 1, Problem 1.1 is completely solved in [8] as follows: For each positive integer n, we have λ V (α) ∼ λ ⊕f n,1(α)= U(gln) · det (X) = Mn , (1.5) λ⊢n fλM(α)6=0 Representation theory of the α-determinant and zonal spherical functions 3 where fλ(α) is a (modified) content polynomial ℓ(λ) λi def fλ(α) = (1 + (j − i)α). i=1 j=1 Y Y In other words, for each λ ⊢ n, we have λ V 0 α ∈{1/k ; 1 ≤ k<ℓ(λ)}⊔{−1/k ; 1 ≤ k < λ1} , multiplicity of Mn in n,1(α)= λ (1.6) (f otherwise. F λ The transition matrix n,1(α) in this case is given by fλ(α)I. F λ Example 1.4. When n = 2, the transition matrix 2,l(α) is of size 1 (i.e. just a polynomial) and it is shown in [4, Theorem 4.1] that V (α) l ∼ (2l−s,s) 2,l(α)= U(gl2) · det (X) = M2 , (1.7) 0≤s≤l (2l−Ms,s) F2,l (α)6=0 where we put (2l−s,s) l−s l F2,l (α)=(1+ α) Gs(α), l (−s) (l − s + 1) (−α)j Gl (α)= j j . s (−l) j! j=0 j X l Here (a)j = Γ(a + j)/Γ(a) is the Pochhammer symbol. We note that Gs(α) is written by a Jacobi polynomial as s − l − 1 −1 Gl (α)= P (−l−1,2l−2s+1)(1+2α). s s F λ In this paper, we show that the entries of the transition matrices n,l(α) are given by a variation of the n spherical Fourier transformation of a certain class function on Snl with respect to the subgroup Sl (Theorem 2.5). This result also provides another proof of Theorem 1.2. Further, we give a new calculation of the (2l−s,s) polynomial F2,l (α) in Example 1.4 by using an explicit formula for the values of zonal spherical functions for the Gelfand pair (S2n, Sn × Sn) due to Bannai and Ito (Theorem 3.1). 2 Irreducible decomposition of Vn,l(α) Fix n,l ∈ N. Consider the standard tableau T with shape (ln) such that the (i, j)-entry of T is (i − 1)l + j. For instance, if n = 3 and l = 2, then 1 2 T = 3 4 . 5 6 We denote by K = R(T) and H = C(T) the row group and column group of the standard tableau T respectively. Namely, K = {g ∈ Snl ; ⌈g(x)⌉ = ⌈x⌉ , x ∈ [nl]} , H = {g ∈ Snl ; g(x) ≡ x (mod l), x ∈ [nl]} , (2.1) where we denote by [nl] the set {1, 2,...,nl}. We put 1 e = k ∈ C[S ]. (2.2) |K| nl kX∈K 4 K. Kimoto This is clearly an idempotent element in C[Snl]. Let ϕ be a class function on H. We put def Φ = ϕ(h)h ∈ C[Snl]. hX∈H Cn ⊗nl C Consider the tensor product space V = ( ) . We notice that V has a (U(gln), [Snl])-module structure given by nl e e def e se-th e Eij · i1 ⊗···⊗ inl = δis,j i1 ⊗···⊗ i ⊗···⊗ inl , s=1 X e e def e e i1 ⊗···⊗ inl · σ = iσ(1) ⊗···⊗ iσ(nl) (σ ∈ Snl) e n Cn where { i}i=1 denotes the standard basis of . The main concern of this section is to solve the Problem 2.1. Describe the irreducible decomposition of the left U(gln)-module V · eΦe.