Representation Theory of the Α-Determinant and Zonal Spherical Functions

REPRESENTATION THEORY OF THE α-DETERMINANT AND ZONAL SPHERICAL FUNCTIONS

KAZUFUMI KIMOTO

Abstract. U · (α) l We investigate the structure of the cyclic module Vn,l(α) = (gln) det (X) by embedding it to the tensor product space (Cn)⊗nl and utilizing the Schur-Weyl duality. We show that the entries of the λ transition matrices Fn,l(α) are given by a variation of the spherical Fourier transformation of a certain class n function on Snl with respect to the subgroup Sl (Theorem 1.4). This result also provides another proof of (2l−s,s) Theorem ??. Further, we calculate the polynomial F2,l (α) by using an explicit formula of the values of zonal spherical functions for the Gelfand pair (S2l, Sl × Sl) due to Bannai and Ito (Theorem 2.1).

1. Irreducible decomposition of Vn,l(α) and transition matrices Let us 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, © ¯ ª (1.1) K = g ∈ S ¯ dg(x)/le = dx/le , x ∈ [nl] , © nl ¯ ª ¯ (1.2) H = g ∈ Snl g(x) ≡ x (mod l), x ∈ [nl] . We put 1 X (1.3) e = k ∈ C[S ]. |K| nl k∈K This is clearly an idempotent element in C[S ]. Let ϕ be a class function on H. We put nl X Φ = ϕ(h)h ∈ C[Snl]. h∈H Cn ⊗nl U C Consider the tensor product space V = ( ) . We introduce a ( (gln), [Snl])-module structure of V by Xnl · ⊗ · · · ⊗ ⊗ · · · ⊗ s-th ⊗ · · · ⊗ Eij ei1 einl = δis,j ei1 ei einl , s=1 ⊗ · · · ⊗ · ⊗ · · · ⊗ ∈ ei1 einl σ = eiσ(1) eiσ(nl) (σ Snl), { }n Cn where ei i=1 denotes the standard basis of . The main concern of this subsection is to describe the irreducible U · decomposition of the left (gln)-module V eΦe. We first show that Vn,l(α) is isomorphic to V · eΦe for a special choice of ϕ. Consider the group isomorphism → l θ : H Sn defined by

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

Date: October 30, 2008. 2000 Mathematics Subject Classiﬁcation. Primary 22E47, 33C45; Secondary 43A90, 13A50. Key words and phrases. Alpha-determinant, cyclic modules, Jacobi polynomials, singly conﬂuent Heun ODE, permanent, Kostka numbers, irreducible decomposition, spherical Fourier transformation, zonal spherical functions, Gelfand pair. 1 2 KAZUFUMI KIMOTO

We also deﬁne an element D(X; ϕ) ∈ A(Matn) by X Yn Yl X Yn Yl D(X; ϕ) = ϕ(h) x = ϕ(h) x −1 θ(h)p(q),q q,θ(h)p (q) h∈H q=1 p=1 h∈H q=1 p=1 X Yn Yl −1 = ϕ(θ (σ1, . . . , σl)) xσp(q),q.

σ1,...,σl∈Sn q=1 p=1 ν(·) (α) l −1 ··· ∈ n We note that D(X; α ) = det (X) since νθ (σ1, . . . , σl) = νσ1 + + νσl for (σ1, . . . , σl) Sl . Take a class function δ on H deﬁned by H ( 1 h = 1 δH (h) = 0 h =6 1.

l We see that D(X; δH ) = (x11x22 . . . xnn) . We need the following lemma (The assertion (1) is just a rewrite of Lemma ??, and (2) is immediate to verify). Lemma 1.1. (1) It holds that U · ⊗l ⊗ · · · ⊗ ⊗l · Sl Cn ⊗n (gln) e1 en = V e = ( ) , M Yn Yl U · C · ∼ Sl Cn ⊗n (gln) D(X; δH ) = xipq q = ( ) .

ipq ∈{1,2,...,n} q=1 p=1 (1≤p≤l, 1≤q≤n) (2) The map

Yn Yl T U · 3 : (gln) D(X; δH ) xipq q q=1 p=1 7−→ ⊗ · · · ⊗ ⊗ · · · ⊗ ⊗ · · · ⊗ · ∈ · (ei11 eil1 ) (ei1n eiln ) e V e U ¤ is a bijective (gln)-intertwiner. We see that T (D(X; ϕ)) Ã ! X Yn Yl T = ϕ(h) xθ(h)p(q),q ∈ hXH q=1 p=1 ⊗ · · · ⊗ ⊗ · · · ⊗ ⊗ · · · ⊗ · = ϕ(h)(eθ(h)1(1) eθ(h)l(1)) (eθ(h)1(n) eθ(h)l(n)) e ∈ h H X ⊗l ⊗ · · · ⊗ ⊗l · · ⊗l ⊗ · · · ⊗ ⊗l · = e1 en ϕ(h)h e = e1 en eΦe h∈H by (2) in Lemma 1.1. Using (1) in Lemma 1.1, we have the Lemma 1.2. It holds that U · ∼ · (gln) D(X; ϕ) = V eΦe U · ∼ ν(h) ¤ as a left (gln)-module. In particular, V eΦe = Vn,l(α) if ϕ(h) = α . By the Schur-Weyl duality, we have M ∼ Mλ £ Sλ V = n . λ`nl λ Here S denotes the irreducible unitary right Snl-module corresponding to λ. We see that ¡ ¢ D E Sλ · G Sλ dim e = indK 1K , = Kλ(ln), Snl REPRESENTATION THEORY OF THE α-DETERMINANT AND ZONAL SPHERICAL FUNCTIONS 3

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 `(λ) ≤ n, it follows the Theorem 1.3. It holds that M ¡ ¢ · ∼ Mλ £ Sλ · V eΦe = n eΦe . λ`nl `(λ)≤n In particular, as a left U(gl )-module, the multiplicity of Mλ in V · eΦe is given by n ¡ ¢ n λ dim S · eΦe = rkEnd(Sλ·e)(eΦe). ¤

` ≤ { λ λ } Let λ nl be a partition such that `(λ) n and put d = Kλ(ln). We fix an orthonormal basis e1 ,..., ef λ Sλ λ λ Sλ K of such that the first d vectors e1 ,..., ed form a subspace ( ) 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 ® λ λ · λ ∈ ≤ ≤ λ (1.4) ψij(g) = ei g, ej Sλ (g Snl, 1 i, j f ). Mλ · We notice that this function is K-biinvariant. We see that the multiplicity of n in V eΦe is given by the rank of the matrix Ã ! X λ ϕ(h)ψij(h) . h∈H 1≤i,j≤d As a particular case, we obtain the Mλ U · (α) l Theorem 1.4. The multiplicity of the irreducible representation n in the cyclic module (gln) det (X) is equal to the rank of Ã ! X λ ν(h) λ (1.5) Fn,l(α) = α ψij(h) , h∈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 (1.4). ¤ λ { λ } Remark 1.5. (1) We have Fn,l(0) = I by the definition of the basis ψij i,j in (1.4). ν(g−1) ν(g) λ −1 λ ∈ λ ∗ (2) Since α = α and ψij(g ) = ψji(g) for any g Snl, the transition matrices satisfy Fn,l(α) = λ Fn,l(α). (3) In Examples 1.6 and 1.8 below, the transition matrices are given by diagonal matrices. We expect that λ C any transition matrix Fn,l(α) is diagonalizable in MatKλ(ln) ( [α]).

Example 1.6. If l = 1, then H = G = Sn and K = {1}. Therefore, for any λ ` n, we have ® λ n! λ (1.6) Fn,1(ϕ) = ϕ, χ I f λ Sn 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 λ (1.7) Fn,1(α) = fλ(α)I ν(·) since the Fourier expansion of α (as a class function on Sn) is X f λ (1.8) αν(·) = f (α)χλ, n! λ λ`n

which is obtained by specializing the Frobenius character formula for Sn (see, e.g. [10]). 4 KAZUFUMI KIMOTO

(nl) S(nl) Example 1.7. Let us calculate Fn,l (α) by using Theorem 1.4. Since is the trivial representation, it follows that (S(nl))K = S(nl) and X X (nl) ν(h) h · i ν(σ1) ν(σl) Fn,l (α) = α e h, e = α . . . α h∈H σ1,...,σl∈Sn = ((1 + α)(1 + 2α) ... (1 + (n − 1)α))l ,

where e denotes a unit vector in S(nl). (nl−1,1) Example 1.8. Let us calculate Fn,l (α) by using Theorem 1.4. As is well known, the irreducible (right) (nl−1,1) nl Snl-module S can be realized in C as follows: ( ¯ ) ¯ ¯ Xnl S(nl−1,1) nl ∈ Cnl ¯ = (xj)j=1 ¯ xj = 0 . j=1 This is a unitary representation with respect to the ordinary hermitian inner product h·, ·i on Cnl. It is immediate to see that ¡ ¢ S(nl−1,1) K n ¯ o ¯ nl ∈ S(nl−1,1) ··· ≤ = (xj)j=1 ¯ xpl+1 = xpl+2 = = x(p+1)l (0 p < n) . ¡ ¢ (nl−1,1) K Take an orthonormal basis e1,..., en−1 of S by

l l l ¡z }| { z }| { z }| {¢ 1 j j 2j 2j nj nj ej = √ ω , . . . , ω , ω , . . . , ω ,..., ω , . . . , ω (1 ≤ j ≤ n − 1), nl (nl−1,1) where ω is a primitive n-th root of unity. Then, the (i, j)-entry of the transition matrix Fn,l (α) is X X Xn Xl 1 − αν(h) he · h, e i = αν(σ1) . . . αν(σl)ωσq (p)i pj i j nl h∈H σ1,...,σl∈Sn p=1 q=1 Ã !l−1 Ã ! X 1 X Xn = αν(τ) αν(σ)ωσ(p)i−pj . n τ∈Sn σ∈Sn p=1 − The ﬁrst factor is ((1 + α)(1 + 2α) ... (1 + (n − 1)α))l 1. We show that

1 X Xn αν(σ)ωσ(p)i−pj = (1 − α)(1 + α)(1 + 2α) ... (1 + (n − 2)α)δ n ij σ∈Sn p=1 (i, j = 1, 2, . . . , n − 1). For this purpose, by comparing the coefficients of αn−m in both sides, it is enough to prove ½· ¸ · ¸¾ 1 X Xn n − 1 n − 1 ωσ(p)i−pj = − δ n m − 1 m ij σ∈Sn p=1 ν(σ)=n−m (i, j, m = 1, 2, . . . , n − 1), £ ¤ n where m denotes the Stirling number of the first kind (see, e.g. [5] for the definition). Since (£ ¤ n−1 x = p, |{σ ∈ S ; ν(σ) = n − m, σ(p) = x}| = £ m−1¤ n n−1 6 m x = p REPRESENTATION THEORY OF THE α-DETERMINANT AND ZONAL SPHERICAL FUNCTIONS 5

for each p, x ∈ [n], it follows that   · ¸ · ¸  1 X Xn 1 Xn n − 1 X n − 1 αν(σ)ωσ(p)i−pj = ω−pj ωpi + ωxi n n  m − 1 m  σ∈Sn p=1 p=1 x=6 p ½· ¸ · ¸¾ ½· ¸ · ¸¾ n − 1 n − 1 1 Xn n − 1 n − 1 = − ωp(i−j) = − δ , m − 1 m n m − 1 m ij p=1 P xi − pi ≤ which is the required conclusion. Here we notice that x=6 p ω = ω since 1 i < n. Consequently, we obtain − F (nl 1,1)(α) n,l ³ ´ l l−1 = (1 − α) ((1 + α)(1 + 2α) ... (1 + (n − 2)α)) (1 + (n − 1)α) δij , 1≤i,j≤n−1 (nl−1,1) so that the multiplicity of Mn in Vn,l(α) is zero if α = −1/k (k = 1, 2, . . . , n − 1) and n − 1 otherwise. The trace of the transition matrix F λ (α) is n,l X λ λ ν(h) λ (1.9) fn,l(α) = tr Fn,l(α) = α ω (h), h∈H where ωλ is the zonal spherical function for λ with respect to K defined by 1 X ωλ(g) = χλ(kg)(g ∈ S ). |K| nl k∈K λ λ This polynomial 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 λ λ −1 λ Sλ K a transition matrix Fn,l(α) is a scalar matrix, then we would have Fn,l(α) = d fn,l(α)I (d = dim( ) ) and Mλ λ hence we see that the multiplicity of n in Vn,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 λ transition matrices are scalar as well as to get an explicit expression for the polynomials fn,l(α). Here we give ` λ a sufficient condition for λ nl such that Fn,l(α) is a scalar matrix. λ Proposition 1.9. (1) Denote by NH (K) the normalizer of K in H. The transition matrix Fn,l(α) is scalar λ K if (S ) is irreducible as a NH (K)-module. − r λ (2) If λ is of hook-type (i.e. λ = (nl r, 1 ) for some r < n), then Fn,l(α) is scalar. ∼ Proof. Notice that N (K) = S . Consider a linear map T ∈ End((Sλ)K ) given by H n Ã ! Xd X ® ν(h) · λ λ ∈ Sλ K T (x) = α x h, ej Sλ ej (x ( ) ), j=1 h∈H λ K λ K where d = dim(S ) . It is direct to check that T gives an intertwiner of (S ) as a NH (K)-module. Hence, λ Sλ K by Schur’s lemma, T is a scalar map (and Fn,l(α) is a scalar matrix) if ( ) is an irreducible NH (K)-module. − r ∼ − r When λ = (nl − r, 1r) for some r < n, it is proved in [2, Proposition 5.3] that (S(nl r,1 ))K = S(n r,1 ) as NH (K)-modules. Thus we have the proposition. ¤ (nl−1,1) (nl−1,1) − Example 1.10. Let us calculate fn,l (α). 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 X X − 1 f (nl 1,1)(α) = αν(h) (fix (kh) − 1) n,l |K| nl h∈H k∈K X 1 X X X = αν(h) δ − αν(h). |K| khx,x h∈H k∈K x∈[nl] h∈H 6 KAZUFUMI KIMOTO

It is easily seen that khx =6 x for any k ∈ K if hx =6 x (x ∈ [nl]). Thus it follows that 1 X X X 1 X 1 δ = δ δ = ﬁx (h)(h ∈ H). |K| khx,x hx,x |K| kx,x l nl k∈K x∈[nl] x∈[nl] k∈K Therefore we have X X − 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 h∈H h∈H nY−2 = (n − 1)(1 − α)(1 − (n − 1)α)l−1 (1 + iα)l. i=1 − − Since the transition matrix F (nl 1,1) is a scalar one and its size is dim S(n−1,1) = n − 1, we get F (nl 1,1)(α) = Q n,l n,l − − − l−1 n−2 l (1 α)(1 (n 1)α) i=1 (1 + iα) In−1 again. λ We will investigate these polynomials fn,l(α) and their generalizations in [?].

2. Irreducible decomposition of V2,l(α) and Jacobi polynomials In this subsection, as a particular example, we consider the case where n = 2 and calculate the transition λ matrix F2,l(α) explicitly. Since the pair (S2l,K) is a Gelfand pair (see, e.g. [10]), it follows that D E S2l Sλ Kλ(l2) = indK 1K , = 1 S2l for each λ ` 2n with `(λ) ≤ 2. Thus, in this case, the transition matrix is just a polynomial and is given by µ ¶ X Xl l (2.1) F λ (α) = tr F λ (α) = αν(h)ωλ(h) = ωλ(g )αs. 2,l 2,l s s h∈H s=0

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 [3, p.218] as p µ ¶µ ¶µ ¶− µ ¶ X 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 where µ ¶ −n, n + α + β + 1, −x Q (x; α, β, N) = F˜ ; 1 n 3 2 α + 1, −N µ ¶µ ¶µ ¶− µ ¶− µ ¶ XN n −n − α − β − 1 −α − 1 1 N 1 x = (−1)j j j j j j j=0 ³ ´ a1,...,ap is the Hahn polynomial (see also [10, p.399]), and n+1F˜n ; x is the hypergeometric polynomial b1,...,bq−1,−N µ ¶ XN j a1, . . . , ap (a1)j ... (ap)j x pF˜q ; x = b , . . . , b − , −N (b1)j ... (bq−1)j(−N)j j! 1 q 1 j=0 for p, q, N ∈ N in general (see [1]). We now re-prove Theorem ?? as follows: Theorem 2.1. Let l be a positive integer. It holds that µ ¶ Xl − 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 for p = 0, 1, . . . , l. REPRESENTATION THEORY OF THE α-DETERMINANT AND ZONAL SPHERICAL FUNCTIONS 7

Proof. Let us put x = −1/α. Then we have µ ¶ Xl l Q (s; l − 1, l − 1, l)αs s p s=0 p µ ¶µ ¶µ ¶− X p 2l − p + 1 l 1 = (−1)j αj(1 + α)l−j j j j j=0 p µ ¶µ ¶µ ¶− X p 2l − p + 1 l 1 = x−l(x − 1)l−p (x − 1)p−j j j j j=0 and p µ ¶µ ¶µ ¶− X 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 Here we use the elementary identity µ ¶µ ¶ µ ¶ Xl l s l αs = αj(1 + α)l−j. s j j s=0 Hence, to prove the theorem, it is enough to verify p µ ¶µ ¶µ ¶− p µ ¶µ ¶µ ¶− X p l − p + i l 1 X p 2l − p + 1 l 1 (2.2) xp−i = (x − 1)p−j. i i i j j j i=0 j=0 Comparing the coeﬃcients of Taylor expansion of these polynomials at x = 1, we notice that the proof is reduced to the equality µ ¶µ ¶ µ ¶ Xr l − i l − p + i 2l − p + 1 (2.3) = l − r l − p r i=0 for 0 ≤ r ≤ p, which is well known (see, e.g. (5.26) in [5]). Hence we have the conclusion. ¤ Thus we obtain the irreducible decomposition M − ∼ M(l,l) ∼ M(2l−p,p) 6 − (2.4) V2,l( 1) = 2 , V2,l(α) = 2 (α = 1) 0≤p≤l l 6 Gp(α)=0 of V2,l(α) again. n Remark 2.2. (1) The calculation above uses the advantage for the fact that the pair (Snl, Sl ) is the Gelfand pair only when n = 2. (2) We have used the result in [3, p.218] for the theorem. It is worth mentioning that one may prove conversely the result in [3, p.218] from Theorem ??. Acknowledgement The author would thank Professor Itaru Terada for noticing that his work [2] is useful for the discussion in Section 6.2. References

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Department of Mathematical Sciences, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan E-mail address: [email protected]