CC BY
32
MSG 16S30, 15A15, 22C30
Representation theory of the a-determinant and zonal spherical functions
Consider the cyclic span of a power of a certain polynomial called a-determinant, which is a common generalization of the determinant and permanent, under the action of the universal enveloping algebra of the general linear algebra. We show that the multiplicity of each irreducible component in this cyclic module is given by the rank of a certain associated
a
also give several explicit examples of such matrices. In particular, in the case where the
a
are essentially given by Jacobi polynomials
Keywords: universal enveloping algebra, determinant, permanent, Young diagram, induced representations, content polynomials, spherical functions
Let us consider the representation of the universal enveloping algebra U(gln) of gln = gIn(C) on the polynomial algebra A(Matn) of n2 variables Xj (1 ^ i, j ^ n) defined by
for f (X) e A(Matn) (X = (xj)i^j,j^n), where {Epq}^p,q^n is the standard basis of gin. It is well known that the cyclic modules generated by the determinant and permanent are both irreducible. Actually, if we denote by Mn the irreducible representation of U(gln) with highest weight A (which we will represent as a partition), then
Namely, det(X) generates the skew-symmetric tensor of the natural representation Cn, and per(X) the symmetric tensor of Cn,
© K. Kimoto
University of the Rvukvus, Nishihara, Okinawa, Japan
1. Introduction
U(0[„) ■ det(X) = MiT, U(0[„) ■ per(X) = Min’.
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As a common generalization of the determinant and permanent, the a-determi-nant is defined by
where v(a) = n — c(^^d c(a) is the number of cycles in the disjoint cycle decomposition of a, In fact, det(X) = det(-1’(X^d per(X) = det(1’(X), It
a
a
a
theory. For further information, we refer to [9], [8] and the references within,
a
symmetric tensor representations, Matsumoto and Wakayama studied the cyclic U(gln)-module generated by the a-determinant and determine the irreducible decomposition of it. Precisely, they proved that
Vni(a) := U(gin) ■ det(a’(X) = ® (M;
\Mn) ,
Ahn
where the multiplicity (a) of the irreducible component Mn is given by
n(a) = { 0A fA(a) 0,
* fA otherwise,
mn(a)
and
^(A) A*
f A(a) = nn(1 +(j — i)a)
i=l j=l
is the (modified) content polynomial for a partition A (£(A) is the length of A), The point is that the irreducible decomposition of the module Vnl (a) is controlled by simple polynomials {fA(a)}Ahn, whose roots are reciprocal of non-zero integers, and the multiplicities are “all-or-nothing” (i.e. the possible values of m^a) is either 0 or fA for each A),
In this article, we consider the generalization
Vni(a) := U(gin) ■ det(a’(X)J•
We will see that the multiplicity of each irreducible representation M^ m Vn1(a) is given by the rank of a certain matrix denoted by (a) (Theorem 2,2), In contrast to the case where l = 1, the multiplicities would take an intermediate value between 0 and the size of the matrix F^a) (see Example 2), and it seems quite difficult so
a
explicit way.
However, we can give a sufficient condition for the matrix F^a) to be .scalar (Proposition 2,1), in which case the multiplicity is controlled by a single polynomial
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fn z(a) = trF^ (a) as in the case of l = 1, One of the most interesting cases of such a scalar situation is the case where n = 2, We will see that f^ t (a) is written in terms of the Jacobi polynomials. As an appendix, we also give several concrete examples of such polynomials.
This article is written based on the talk given at the workshop “Harmonic Analysis on Homogeneous Spaces and Quantization” (February 18-22, 2008) in Fukuoka as well as our recent article [3], which is a joint work with Sho Matsumoto and Masato Wakayama, We will not give proofs of the statements, which one can find in [3].
The author thanks Itaru Terada for letting him know the work [1], which is used to give Example 2,7, after the talk at the workshop. He also thank Masashi Kosuda for the help in the calculation by MAPLE,
2. Irreducible decomposition of Vn1 (a) and transition
matrices
Fix positive integers n, l. Take a standard tableau T with shape (ln) (i, e, a rectangular tableau with n rows and l columns) such that the (i, j)-entry of T is (i — 1)l + j, and denote by K = Row(T) and H = Col(T) the row group and column T
e := |jK| ^ k $ := ^ ^(h)h
1 1 fceK
in the group algebra C[Sn1], where ^ is a class function on H, play a key role. We will work on the tensor product space V = (Cn)®n1, which is a (U(gln), C[Sn1])-module by setting
nl
í „• e. & •••(£) e & •••(£) e .,
(2.2)
Ey ■ Si, 0 ■ ■ ■ 0 e,nl = ^ í,, , j e,, 0 ■ ■ ■ 0 e, 0 ■ ■ ■ 0 e,nl
s=l
eii 0 ■■ ■ 0 einl ■ a = eiCT(1) 0 ■ ■ ■ 0 eiff(nI) (a G 6ni),
where {e^n^ denotes the standard basis of Cn, in the first formula ei stands at the s-th place. Using the group isomorphism
0 : H 9 h ^ 0(h) = (0(h)i,..., 0(h),) G Sn,
0(h)j(x) = y ^ h((x — 1)l + j) = (y — 1)l + j
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where 1 ^ x,y ^ n 1 ^ j ^ h define also an element D(X; p) e A(Matn) by
n 1
D(X; p)=£ PWlIII x0(h)p (q),q
h€H q=1 p=1
n 1
= ^ ^(rVl,...,^)) XCTp(q),q. (2.4)
Cl ffi£6„ q=1 p=1
Notice that D(X; av(^) = det(a)(X) since v(0 1(a1,... , a,)) = v(a1) + ... + v(a,) for (a1,..., a,) e Sn If is a function on H which is one at the identity element and zero otherwise, then D(X; ) = (x11x22 ...xnn)1. The following lemma is
fundamental.
Lemma 2.1 (1) U(gln) ■ ef1 ® ■ ■ ■ ® ef1 = V ■ e = S 1(Cn)®n.
(2) Tfte map T : U(gln) ■ D(X; ^ V ■ e (/¿fen
n 1
nn Xi(p,q),q I > (ei(1,1) 0 ■ ■ ■ 0 ei(1,1)) 0 ■ ■ ■ 0 (ei(1,n) 0 ■ ■ ■ 0 ei(1,n)) ■ e
q=1p=1
where i(p,q) = ipq = ap(q), see (2-4), is a bijeetive U(gln)-intertwiner.
(3) For any class function p on H, the polynomial D(X; p) belongs to
U (gln) ■ D(X; ) and, is mapped to ef1 0---0 ef1 ■ e$e by T.
Using the lemma, we have the
Lemma 2.2 It holds that
U(gin) ■ D(X; p) = V ■ e$e (2.5)
as a left U(Qln)-module. In particular, V ■ e$e = Vn1 (a) if p(h) = av(h).
The Schur-Weyl duality reads
V = © Mn B SA. (2.6)
Ahnl
Here SA denotes the irreducible unitarv right Sn1-module corresponding to A. We see that
where 1K is the trivial representation of K and (n,p)S ; is the intertwining number of given representations n and p of Sn^^d KAjU is the Kostka number. Since KA(p) = 0 unless £(A) ^ n, it follows the
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Theorem 2.3 The irreducible decomposition
(2.8)
Abnl
¿(A)<n
holds, so that the multiplicity of in V ■ e$e is (/¿fen by
dim (SA ■ e$e) = rankEnd(SA• e)(e$e).
(2.9)
As a special case, we now obtain the
Theorem 2.4 Let d = KA(,«). Fix an orthonormaI basis {e^}^ of SA, and denote by {^A-} the matrw coefficients relative to this basis. Su,ppose that the first d vectors eA,..., e^ spans (Sa)K. Then the multiplicity of the irreducible representation Mn m ifte cyclic module U(gln) ■ det(a)(X) ¿s egwal to the rank of the matrix
f£,(a) := (h)
(2.10)
We refer to the matrix F^ (a) as a transition matrix for A. The transition matrix itself does depend on the choice of the basis {eA}^ of SA in the theorem, while its rank does not. The trace of the transition matrix FA1(a) is
where wA is the zonal spherical function for A with respect to K defined by
If the matrix F^(a) is scalar, then FA1(a) = d-1/A ,(a) Id (recall d = dim(SA)K) and hence the multiplicity of in Vn1 (a) is completely controlled by the single polynomial /A ,(a) as in the case where l = 1. Thus it is desirable to obtain a characterization of the triplets (n, l, A) such that FA ¿(a) are scalar. The following is a sufficient condition for A when n and l are given.
Proposition 2.5 Denote by (K) the normalizer of K in H ( Notice that (K) = Sn ). The transition matrix FA,(a) is scalar if (SA)K ¿s irreducible as a (K )-module.
Example 2.6 (Matsumoto-Wakayama case). If l = 1, then K = 1 and (K) = H so that (SA)K = SA is an irreducible (K)-module, and hence all the transition matrices F^ 1 (a) are scalar. In fact, we have F^ 1(a) = /A(a)I and /A 1(a) = /A/A(a).
(2.11)
(2.12)
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Example 2.7 (hook-type case). If A = (nl — r, 1r) is of hook type (0 ^ r ^ n — 1), then (S(n1-r,ir))K = S(n-r,ir) as (K)-modules by [1, Proposition 5,3], Thus the transition matrix Fin-7"’1 )(a) is scalar. See Appendix for the concrete examples in this case.
Example 2.8 (Gelfand pair case). Suppose that (Snl,K) is a Gelfand pair, that is, the induced representation IndK«1K of the trivial representation 1K of K to &ni is multiplicity-free (see, e.g. [6]), Then (SA)K is obviously irreducible as an (K)-module since it is one-dimensional. In this case, each transition matrix is just a polynomial (one by one matrix). We give an explicit formula of the transition matrices for the case where n = 2 in the next section.
We also give a non-scalar example of a transition matrix.
Example 2.9 Taking a suitable orthonormal basis of (S(4,2)) , we have the transition matrix F3(42’2)(a) for M3422)(a) in V3,2(a) as
F3(42’2)(a) = ^(1 + a)2diag{2 — 2a + 3a2,1 — a, 1 — a}.
Hence, the multiplicity mif^a) of M3422)(a) in V3,2(a) is
(4,2) / \
m3,2 (a) =
0 a = -1,
1 a =1,
2 a = (1 ±V-5)/3,
3 otherwise.
3. Irreducible decomposition of V2,/(a) and Jacobi
polynomials
When n = 2, as is well known, the pair (S2l,K) is a Gelfand pair, so that the transition matrices F^a), where A h 2l, £(A) ^ 2, are scalar (of size one). If we set = (1, l + 1)(2, l + 2)... (s, l + s) £ ©2n; then we have
Now we write A = (2l — r, r) for some r (0 ^ r ^ l). The value w(21 r’r)(gs) of the zonal spherical function is calculated by Bannai and Ito [2, p, 218] as
c(2l-r,r)(gs) = Qr(s; -/ - 1, -/ - 1,/)
r
= t<-D
^ , r\ /2/ - r + 1\ / A Vs
j=0 vV V J / VJ/ VJ
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where
is the Hahn polynomial (see also [6, p. 399]),
l
l
F<'2l-r'r)(a) = £ H Qr (s; l — 1, l - 1, l) as
s=0 W
= ^ ^ ^ (1 + a)l-r Pr(-1-1’2l-2r+1) (1 + 2a)
for r = 0,1,..., l. Here Pn“’b) (x) denotes the Jacobi polynomial
P.“’6)(x) = (n + “) 2F1 (—n, a + b + n +1, a +1; X) .
Further, all roots of F2(2ll-r’r) (a) are lying on the unit circle |z| = 1.
Thus we obtain the irreducible decomposition of V2,l(a):
V2,l(—1) = M2l,l),
V2’l(a) — M221 a = — ^-,
the sum is taken over r = 0,1,..., l, such that P-(-1-1’2l-2r+1)(1 + 2a) = 0,
§ 4. Remarks on related works
For all but finite values of a, Vnl(a) is equivalent to Sl(Cn)®n as we see above. It is interesting not only to describe the exceptional singular values nicely (as zeros of certain special polynomials, for instance), but also to investigate what happens at the singular values.
We study the quantum analogue of our problem in [5] from the first point of a
matrix algebra, and consider the cyclic span generated by it under the action of the quantum enveloping algebra. What we expect in this direction is to obtain certain special polynomials in a with parameter q as (entries of) transition matrices defined analogously,
a
a negative integer. In this case, the — 1/fc-determinants satisfies a “ — 1 /fc-analogue”
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of the multiplicativity of the determinant (k = 1, 2,..., n — 1, n being the size of the matrix). This enables us to construct a certain relative invariant of GLn, which we call the wreath determinants, using the — 1/k-determinant, It would be interesting to explore wreath analogues of various known determinant formulas. As an example of such ones, we give an analogue of the Cauchy determinant formula (see § 6 of [4]),
Appendix. Examples of traces of the transition matrices for hook-types
Here we give several examples of the trace /A l (a) = trFA l(a) of the transition matrices for the case where A is of hook-tvpe calculated by MAPLE, First remark that we can calculate /Al(a) explicitly for A = (nl), (nl — 1,1) as follows:
n—1
= AW = n(1+ja)l, j=1
/i?—1 ’1)(a) = /inn1)(a)l—1/inn1—1,1)(a)
n— 2
= (n — 1)(1 — a)(1 — (n — 1)a)l—1 n(1+ ja)l.
j=1
Here are some other examples:
• (n,l) = (5, 2):
2 11 /5,2 = 6(1 + a)2(1 + 2a)2(1 + 3a)(1 — a)(1 + 2a — — a2),
3 9
/5,;2 = 4(1 + a)2(1 + 2a)(1 + 3a)(1 — a)(1 — 2a)(1 + a — - a2),
/56;214 = (1 + a)(1 + 2a)2(1 — a)(1 — 2a)2(1 — 6a2).
• (n,l) = (4, 2) :
/46;212 = 3(1 + a)2(1 + 2a)(1 — a)(1 + a — 4a2),
/45;213 = (1 + a)(1 + 2a)(1 — a)(1 — 2a)(1 — 3a2).
• (n, l) = (3, 2):
25 /3,2 =(1 + a)(1 — a)(1 — 2 a2).
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• (n,l) = (4, 3) :
2 10
/51 = 3(1 + a)3(1 + 2a)2(1 + 3a)(1 — a)(1 + a-a2),
4,3 3
/49,313 = (1 + a)2(1 + 2a)2(1 — a)(1 + a — 7a2 — 17 a3 + 94 a4).
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• (n,l) = (3, 3) :
/"li = (1 + a)2(1 +2a)(1 — a)(1 — 2a2).
•(n,l) = (3,4):
27
/ff =(1 + a)3(1 + 2a)2(1 — a)(1 — 7 a2).
References
1, S, Ariki, J, Matsuzawa and I, Terada, Representation of Weyl groups on zero weight spaces of g-modules, Algebraic and topological theories (Kinosaki, 1984), Kinokuniva, Tokyo, 1986, 546-568,
2, E, Bannai and T, Ito, Algebraic Combinatorics I, Association Schemes, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984,
3, K, Kimoto, S, Matsumoto and M, Wakayama, Alpha-determinant cyclic modules and Jacobi polynomials, to appear in Trans, Amer, Math, Soc,
a
J, Combin, Theory Ser, A, 2008, vol. 115, No, 1, 1-31,
a
Uq(gin), J. Algebra, 2007, vol. 313, 922-956.
6,1. G. Macdonald. Symmetric Functions and Hall Polynomials, 2nd edn,, Oxford University Press, 1995,
7, S, Matsumoto and M, Wakayama, Alpha-determinant cyclic modules of gin(C),
8, T, Shirai and Y, Takahashi, Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes, J, Funct, Anal., 2003, vol. 205, 414-463.
9, D. Vere-Jones, A generalization of permanents and determinants, Linear Algebra Appl,, 1988, vol. Ill, 119-124,
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