arXiv:0712.2293v1 [math.RT] 14 Dec 2007 Let Introduction 1 ucinon function h eemnn n permanent. and determinant the ent htapaa nlgeo h eeJnsiett 12 ha (1.2) cert identity a Vere-Jones define the to of it analogue use pfaffian and a operator that integral note class [10 We trace Takahashi a tr and unified for Shirai a Further, (1.2) supplies distributions. (1.2) identity binomial the negative Actually, theory. probability to where endby defined define per( eraiyseta the that see readily We o an for I eoe h dniymti fsial ie i neto ftestud the of intention His size. suitable of matrix identity the denotes Let The n A when loanwapc fterpeetto hoyo the of theory representation the of aspect new a also Keywords: 00Mteaia ujc Classification: Subject Mathematical 2000 polynomials. Jacobi functions, aito fteshrclFuirtasomto o ( for transformation Fourier spherical the of variation the )of )) eapstv nee.W eoeby denote We integer. positive a be m n α α i by epoeta h utpiiyo ahirdcbecomponen irreducible each of multiplicity the that prove We l ( dtriati rtitoue yVr-oe 1] epoe the proved He [11]. Vere-Jones by introduced first is -determinant eacmlxnme and number complex a be t oe det power -th σ A n stenme of number the is ) n S ersnainter fthe of theory Representation .Ti ie o nyaohrpofo h eutb Kimoto- by result the of proof another only not gives This 2. = when n matrix ti ayt e ht( that see to easy is It . lh-eemnn,cci oue,irdcbedecompos irreducible modules, cyclic Alpha-determinant, α A = ( α det( ) ( = − ( α X (resp. 1 n oa peia functions spherical zonal and dtriatdet -determinant a ) I l ij − fthe of ) i 1 αA cce ntedson yl eopsto of decomposition cycle disjoint the in -cycles ≤ det i,j ) α ≤ α − ( ν α A n dtriati ie ytern famti hs nre ar entries whose matrix a of rank the by given is -determinant ) ec ergr the regard we Hence 1). = 1 ( ) /α σ ( uhta h bouevleo n ievleof eigenvalue any of value absolute the that such − ( = A ) 1) = def ) = S a ν def = auuiKIMOTO Kazufumi k X ( ij ∞ eebr1,2007 14, December n X ( =0 i σ α ≥ ) ) σ h ymti ru fdegree of group symmetric the ) 1 1 X ( ∈ sgn = k ≤ ( 1 A S i ! i,j onie ihtedtriatdet( determinant the with coincides ) n − 1 Abstract ≤ ≤ α 24,43A90. 22E47, 1) n i ν 1 σ X ,...,i ( m an σ 1 α stesgaueo permutation a of signature the is S ) i a -determinants. ( k nl n σ σ ≤ (1)1 , ( ) n by S det l n a .Frhr ecluaetemti explicitly matrix the calculate we Further, ). n σ ( (2)2 α α nthe in t arx The matrix. σ i n-aaee aiyo on processes. point of family one-parameter ain ) dtriata omngnrlzto of generalization common a as -determinant    rvdaFehl eemnn eso of version determinant Fredholm a proved ] ∈ a . . . a a amn ftemliait ioiland binomial multivariate the of eatment enas salse n sapidto applied is and established also been s S i i α k 1 . . . fthe of y i i n to,Gladpi,znlspherical zonal pair, Gelfand ition, 1 1 σ -determinant ) U asmt-aaaa(07 but (2007) Matsumoto-Wakayama ( , n ( gl a . . . a . . . ) . n identity . n n . . σ -ylcmdl eeae by generated module )-cyclic α o permutation a For . α entc that notice We . dtriatdet -determinant dtriati napplication an is -determinant i i k 1 . . . i i k k    A A σ rs.permanent (resp. ) . sls hn1 Here 1. than less is ie ya by given e ( ν α ( ) · σ ( saclass a is ) A ∈ of ) S n (1.1) (1.2) A we , is 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. Here we show that Problem 2.1 includes Problem 1.1 as a special case. We consider the group isomorphism l θ : H → Sn defined by

def θ(h) = (θ(h)1,...,θ(h)l); θ(h)i(x)= y ⇐⇒ h((x − 1)l + i) = (y − 1)l + i.

We also define an element D(X; ϕ) ∈P(Matn) by

n l n l def D(X; ϕ) = ϕ(h) x = ϕ(h) x −1 θ(h)p(q),q q,θ(h)p (q) q=1 p=1 q=1 p=1 hX∈H Y Y hX∈H Y Y n l −1 = ϕ(θ (σ1,...,σl)) xσp(q),q. S q=1 p=1 σ1,...,σXl∈ n Y Y ν(·) (α) l −1 n We note that D(X; α ) = det (X) since ν(θ (σ1,...,σl)) = ν(σ1)+ ··· + ν(σl) for (σ1,...,σl) ∈ Sl . Take a class function δH on H defined by

1 h =1 δH (h)= (0 h 6=1.

l We see that D(X; δH ) = (x11x22 ...xnn) . We need the following lemma (see [4, Lemma 2.1] for the proof of (1). The assertion (2) is immediate).

Lemma 2.2. (1) It holds that

e⊗l e⊗l l Cn ⊗n U(gln) · 1 ⊗···⊗ n = V · e = Sym ( ) , n l C ∼ l Cn ⊗n U(gln) · D(X; δH )= · xipq q = Sym ( ) .

ipq ∈{1,2,...,n} q=1 p=1 (1≤p≤Ml, 1≤q≤n) Y Y

(2) The map

n l e e e e T : U(gln) · D(X; δH ) ∋ xipq q 7−→ ( i11 ⊗···⊗ il1 ) ⊗···⊗ ( i1n ⊗···⊗ iln ) · e ∈ V · e q=1 p=1 Y Y is a bijective U(gln)-intertwiner. Representation theory of the α-determinant and zonal spherical functions 5

We see that

n l

T (D(X; ϕ)) = ϕ(h)T xθ(h)p(q),q q=1 p=1 ! hX∈H Y Y e e e e = ϕ(h)( θ(h)1(1) ⊗···⊗ θ(h)l(1)) ⊗···⊗ ( θ(h)1(n) ⊗···⊗ θ(h)l(n)) · e hX∈H e⊗l e⊗l e⊗l e⊗l = 1 ⊗···⊗ n · ϕ(h)h · e = 1 ⊗···⊗ n · eΦe hX∈H by (2) in Lemma 2.2. Using (1) in Lemma 2.2, we have the Lemma 2.3. It holds that ∼ U(gln) · D(X; ϕ) = V · eΦe

∼ V ν(h) as a left U(gln)-module. In particular, V · eΦe = n,l(α) if ϕ(h)= α . By the Schur-Weyl duality, we have

∼ λ ⊠ λ V = Mn S . λM⊢nl λ Here S denotes the irreducible unitary right Snl-module corresponding to λ. We see that

λ G λ dim S · e = indK 1K , S = Kλ(ln) Snl
D E where 1 is the trivial representation of K and hπ, ρi is the intertwining number of given representations π K Snl and ρ of Snl. Since Kλ(ln) = 0 unless ℓ(λ) ≤ n, it follows the Theorem 2.4. It holds that

∼ λ ⊠ λ V · eΦe = Mn S · eΦe . λ⊢nl ℓ(Mλ)≤n

λ In particular, as a left U(gln)-module, the multiplicity of Mn in V · eΦe is given by

λ dim S · eΦe = rkEnd(Sλ·e)(eΦe).

eλ eλ Let λ ⊢ nl be a partition such that ℓ(λ) ≤ n and put d = Kλ(ln). We fix an orthonormal basis { 1 ,..., f λ } λ eλ eλ λ K of S such that the first d vectors 1 ,..., d form a subspace (S ) consisting of K-invariant vectors and left λ λ K f −d vectors form the orthocomplement of (S ) with respect to the Snl-invariant inner product. The matrix coefficient of Sλ relative to this basis is

ψλ (g)= eλ · g, eλ (g ∈ S , 1 ≤ i, j ≤ f λ). (2.3) ij i j Sλ nl

λ We notice that this function is K-biinvariant. We see that the multiplicity of Mn in V · eΦe is given by the rank of the matrix

λ ϕ(h)ψij (h) . ! hX∈H 1≤i,j≤d As a particular case, we obtain the 6 K. Kimoto

λ (α) l Theorem 2.5. The multiplicity of the irreducible representation Mn in the cyclic module U(gln) · det (X) is equal to the rank of F λ ν(h) λ n,l(α)= α ψij (h) , (2.4) ! hX∈H 1≤i,j≤d λ where {ψij }i,j denotes a basis of the λ-component of the space C(K\Snl/K) of K-biinvariant functions on Snl given by (2.3). λ F λ Remark 2.6. (1) By the definition of the basis {ψij }i,j in (2.3), we have n,l(0) = I.

− ν(g 1 ) ν(g) λ −1 λ F λ ∗ (2) Since α = α and ψij (g ) = ψji(g) for any g ∈ Snl, the transition matrices satisfy n,l(α) = F λ n,l(α). (3) In Examples 1.3 and 1.4, the transition matrices are given by diagonal matrices. We expect that any F λ C transition matrix n,l(α) is diagonalizable in MatKλ(ln) ( [α]).

Example 2.7 (Example 1.3). If l = 1, then H = G = Sn and K = {1}. Therefore, for any λ ⊢ n, we have

F λ n! λ n,1(ϕ)= ϕ, χ S I (2.5) f λ n

λ by the orthogonality of the matrix coefficients. Here χ denotes the irreducible character of Sn corresponding to λ. In particular, if ϕ = αν(·), then F λ n,1(α)= fλ(α)I (2.6) ν(·) since the Fourier expansion of α (as a class function on Sn) is

f λ αν(·) = f (α)χλ, (2.7) n! λ λX⊢n which is obtained by specializing the Frobenius character formula for Sn (see, e.g. [6]). F λ The trace of the transition matrix n,l(α) is

λ def F λ ν(h) λ Fn,l(α) = tr n,l(α)= α ω (h), (2.8) hX∈H where ωλ is the zonal spherical function for λ with respect to K defined by 1 ωλ(g) def= χλ(kg) (g ∈ S ). |K| nl kX∈K λ λ This is regarded as a generalization of the modified content polynomial since Fn,1(α)= f fλ(α) as we see above. It is much easier to handle these polynomials than the transition matrices. If we could prove that a transition F λ F λ −1 λ matrix n,l is a scalar matrix, then we would have n,l = d Fn,l(α)I and hence we see that the multiplicity of λ V λ Mn in n,l(α) is completely controlled by the single polynomial Fn,l(α). In this sense, it is desirable to obtain a characterization of the irreducible representations whose corresponding transition matrices are scalar as well λ λ as to get an explicit expression for the polynomials Fn,l(α). We will investigate these polynomials Fn,l(α) and their generalizations in [3].

(nl−1,1) (nl−1,1) Example 2.8. Let us calculate Fn,l (α). We notice that χ (g)=fixnl(g) − 1 where fixnl denotes the number of fixed points in the natural action Snl y [nl]. Hence we see that 1 1 F (nl−1,1)(α)= αν(h) (fix (kh) − 1) = αν(h) δ − αν(h). n,l |K| nl |K| khx,x hX∈H kX∈K hX∈H kX∈K xX∈[nl] hX∈H Representation theory of the α-determinant and zonal spherical functions 7

It is easily seen that khx 6= x for any k ∈ K if hx 6= x (x ∈ [nl]). Thus it follows that 1 1 1 δ = δ δ = fix (h) (h ∈ H). |K| khx,x hx,x |K| kx,x l nl kX∈K xX∈[nl] xX∈[nl] kX∈K Therefore we have 1 F (nl−1,1)(α)= αν(h) fix (h) − αν(h) = F (n)(α)l−1F (n−1,1)(α) n,l l nl n,1 n,1 hX∈H hX∈H n−2 = (n − 1)(1 − α)(1 − (n − 1)α)l−1 (1 + iα)l. i=1 Y F (nl−1,1) (nl−1,1) We note that the transition matrix n,l is a scalar one (see [4]), so that the multiplicity of Mn in Vn,l(α) is zero if α ∈{1, −1, −1/2,..., −1/(n − 1)} and n − 1 otherwise.

3 Irreducible decomposition of V2,l(α) and Jacobi polynomials In this section, as a particular example, we consider the case where n = 2 and calculate the transition matrix F λ 2,l(α) explicitly. Since the pair (S2l,K) is a Gelfand pair (see, e.g. [6]), it follows that

S2l λ Kλ(l2) = indK 1K , S =1 S2l D E for each λ ⊢ 2n with ℓ(λ) ≤ 2. Thus, in this case, the transition matrix is just a polynomial and is given by

l l F λ (α)=tr F λ (α)= αν(h)ωλ(h)= ωλ(g )αs. (3.1) 2,l 2,l s s s=0 hX∈H X   Here we put gs = (1,l + 1)(2,l + 2) ... (s,l + s) ∈ S2n. Now we write λ = (2l − p,p) for some p (0 ≤ p ≤ l). (2l−p,p) The value ω (gs) of the zonal spherical function is calculated by Bannai and Ito [2, p.218] as p p 2l − p +1 l −2 s ω(2l−p,p)(g )= Q (s; −l − 1, −l − 1,l)= (−1)j , s p j j j j j=0 X       where −n,n + α + β +1, −x Q (x; α,β,N) def= F˜ ;1 n 3 2 α +1, −N   N n −n − α − β − 1 −α − 1 −1 N −1 x = (−1)j j j j j j j=0 X        

a1,...,ap is the Hahn polynomial (see also [6, p.399]). We also denote by n+1F˜n ; x the hypergeometric b1,...,bq−1,−N polynomial  

a ,...,a N (a ) ... (a ) xj F˜ 1 p ; x = 1 j p j p q b ,...,b , −N (b ) ... (b ) (−N) j! 1 q−1 j=0 1 j q−1 j j   X for p,q,N ∈ N in general (see [1]). Further, if we put

p −p,l − p +1 p l − p + j l −1 Gl (x) def= F˜ ; −x = (−1)j xj , p 2 1 −l j j j j=0   X     then we have the 8 K. Kimoto

Theorem 3.1. Let l be a positive integer. It holds that

l l F (2l−p,p)(α)= Q (s; l − 1,l − 1,l)αs =(1+ α)l−pGl (α) 2,l s p p s=0 X   for p =0, 1,...,l. Proof. Let us put x = −1/α. Then we have

l p l p 2l − p +1 l −1 Q (s; l − 1,l − 1,l)αs = (−1)j αj (1 + α)l−j s p j j j s=0 j=0 X   X     p p 2l − p +1 l −1 = x−l(x − 1)l−p (x − 1)p−j j j j j=0 X     and p p l − p + j l −1 (1 + α)l−pGl (α)= x−l(x − 1)l−p (−1)j (−x)p−j . p j j j j=0 X     Here we use the elementary identity

l l s l αs = αj (1 + α)l−j . s j j s=0 X      Hence, to prove the theorem, it is enough to verify

p p p l − p + i l −1 p 2l − p +1 l −1 xp−i = (x − 1)p−j . (3.2) i i i j j j i=0 j=0 X     X     Comparing the coefficients of Taylor expansion of these polynomials at x = 1, we notice that the proof is reduced to the equality r l − i l − p + i 2l − p +1 = (3.3) l − r l − p r i=0 X      for 0 ≤ r ≤ p, which is well known (see, e.g. (5.26) in [9]). Thus we have the conclusion. Thus we give another proof of the irreducible decomposition (1.7).

References

[1] G. E. Andrews, R. Askey and R. Roy: Special Functions. Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. [2] E. Bannai and T. Ito: Algebraic Combinatorics I, Association Schemes. The Benjamin/Cummings Publish- ing Co., Inc., Menlo Park, CA, 1984. [3] K. Kimoto: Generalized content polynomials toward α-determinant cyclic modules. Preprint (2007). [4] K. Kimoto, S. Matsumoto and M. Wakayama: Alpha-determinant cyclic modules and Jacobi polynomials. arXiv: 0710.3669. [5] K. Kimoto and M. Wakayama: Invariant theory for singular α-determinants. J. Combin. Theory Ser. A 115 (2008), no.1, 1–31. Representation theory of the α-determinant and zonal spherical functions 9

[6] I. G. Macdonald: Symmetric Functions and Hall Polynomials, Second edition. Oxford Univ. Press, 1995. [7] S. Matsumoto: Alpha-pfaffian, pfaffian point process and shifted Schur measure. Linear Algebra Appl. 403 (2005), 369–398. C [8] S. Matsumoto and M. Wakayama: Alpha-determinant cyclic modules of gln( ). J. Lie Theory 16 (2006), 393-405. [9] R. L. Graham, D. E. Knuth and O. Patashnik: Concrete Mathematics. A foundation for computer science. Second edition. Addison-Wesley Publishing Company, Reading, MA, 1994. [10] T. Shirai and Y. Takahashi: Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes. J. Funct. Anal. 205 (2003), 414–463. [11] D. Vere-Jones: A generalization of permanents and determinants. Linear Algebra Appl. 111 (1988), 119– 124.

Kazufumi KIMOTO Department of Mathematical Science, University of the Ryukyus Senbaru, Nishihara, Okinawa 903-0231, Japan [email protected]