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# Теория представлений для -детерминанта и зональные сферические функции.

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```Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
MSC 16S30, 15A15, 22C30
Representation theory of the ?-determinant
and zonal spherical functions
c
K. Kimoto
University of the Ryukyus, Nishihara, Okinawa, Japan
Consider the cyclic span of a power of a certain polynomial called ?-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
matrix called transition matrices, whose entries are polynomials in the parameter ?. We
also give several explicit examples of such matrices. In particular, in the case where the
size of the matrix for the ?-determinant is two, the polynomials in the transition matrices
are essentially given by Jacobi polynomials
Keywords:
universal enveloping algebra, determinant, permanent, Young diagram, induced
representations, content polynomials, spherical functions
џ 1. Introduction
Let us consider the representation of the universal enveloping algebra U(gln ) of
gln = gln (C) on the polynomial algebra A(Matn ) of n2 variables xij (1 6 i, j 6 n)
dened by
Epq · f (X) =
n
X
r=1
xpr
?f (X)
?xqr
for f (X) ? A(Matn ) (X = (xij )16i,j6n ), where {Epq }16p,q6n is the standard basis
of gln . It is well known that the cyclic modules generated by the determinant
and permanent are both irreducible. Actually, if we denote by M?n the irreducible
representation of U(gln ) with highest weight ? (which we will represent as a
partition), then
n
U(gln ) · det(X) = Mn(1 ) ,
U(gln ) · per(X) = M(n)
n .
Namely, det(X) generates the skew-symmetric tensor of the natural representation
Cn , and per(X) the symmetric tensor of Cn .
1690
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
As a common generalization of the determinant and permanent, the ?-determinant is dened by
X
det(?) (X) =
??(?) x?(1)1 x?(2)2 . . . x?(n)n ,
??Sn
where ?(?) = n ? c(?) and c(?) is the number of cycles in the disjoint cycle
decomposition of ? . In fact, det(X) = det(?1) (X) and per(X) = det(1) (X). It
was Vere-Jones [9] who introduce the ?-determinant rst (but his denition is a
little bit dierent from ours and he also called his one ?-permanent). One of
his motivation to introduce the ?-determinant is an application to the probability
theory. For further information, we refer to [9], [8] and the references within.
Regarding that the ?-determinant interpolates the skew-symmetric tensor and
symmetric tensor representations, Matsumoto and Wakayama studied the cyclic
U(gln )-module generated by the ?-determinant and determine the irreducible
decomposition of it. Precisely, they proved that
M
?m?n (?)
Vn1 (?) := U(gln ) · det(?) (X) =
M?n
,
?`n
where the multiplicity m?n (?) of the irreducible component M?n is given by
(
0
f? (?) = 0,
?
mn (?) =
?
f
otherwise,
and
`(?) ?i
Y
Y
(1 + (j ? i)?)
f (?) =
?
i=1 j=1
is the (modied) content polynomial for a partition ? (`(?) is the length of ?). The
point is that the irreducible decomposition of the module Vn1 (?) is controlled by
simple polynomials {f? (?)}?`n , whose roots are reciprocal of non-zero integers, and
the multiplicities are all-or-nothing (i.e. the possible values of m?n (?) is either 0 or
f ? for each ?).
Vnl (?) := U(gln ) · det(?) (X)l .
We will see that the multiplicity of each irreducible representation M?n in Vnl (?) is
?
given by the rank of a certain matrix denoted by Fn,l
(?) (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 Fn,l
(?) (see Example 2), and it seems quite dicult so
far to determine the exact values of the multiplicities for a given value of ? in an
explicit way.
?
However, we can give a sucient condition for the matrix Fn,l
(?) to be scalar
(Proposition 2.1), in which case the multiplicity is controlled by a single polynomial
1691
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
?
?
(?) as in the case of l = 1. One of the most interesting cases of such a
(?) = trFn,l
fn,l
?
scalar situation is the case where n = 2. We will see that fn,l
(?) is written in terms
of the Jacobi polynomials. As an appendix, we also give several concrete examples
of such polynomials.
Analysis on Homogeneous Spaces and Quantization (February 1822, 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
nd 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.
џ 2. Irreducible decomposition of Vn,l (?) 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
group of T respectively. The following two elements
e :=
1 X
k,
|K| k?K
? :=
X
(2.1)
?(h)h
h?H
in the group algebra C[Snl ], where ? is a class function on H , play a key role. We will
work on the tensor product space V = (Cn )?nl , which is a (U(gln ), C[Snl ])-module
by setting
Eij · ei1 ? · · · ? einl =
nl
X
?is , j ei1 ? · · · ? ei ? · · · ? einl ,
s=1
ei1 ? · · · ? einl · ? = ei?(1) ? · · · ? ei?(nl)
(2.2)
(? ? Snl ),
where {ei }ni=1 denotes the standard basis of Cn , in the rst formula ei stands at the
s-th place. Using the group isomorphism
? : H 3 h 7? ?(h) = (?(h)1 , . . . , ?(h)l ) ? Sln ,
?(h)j (x) = y ?? h((x ? 1)l + j) = (y ? 1)l + j,
1692
(2.3)
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
where 1 6 x, y 6 n, 1 6 j 6 l, dene also an element D(X; ?) ? A(Matn ) by
D(X; ?) =
X
?(h)
x?(h)p (q),q
q=1 p=1
h?H
X
=
n Y
l
Y
?(??1 (?1 , . . . , ?l ))
?1 ,...,?l ?Sn
n Y
l
Y
(2.4)
x?p (q),q .
q=1 p=1
Notice that D(X; ??(·) ) = det(?) (X)l since ?(??1 (?1 , . . . , ?l )) = ?(?1 ) + . . . + ?(?l )
for (?1 , . . . , ?l ) ? Snl . If ?H is a function on H which is one at the identity element
and zero otherwise, then D(X; ?H ) = (x11 x22 . . . xnn )l . The following lemma is
fundamental.
Lemma 2.1
(2)
l
n ?n
?l
.
(1) U(gln ) · e?l
1 ? · · · ? en = V · e = S (C )
The map
n Y
l
Y
T : U(gln ) · D(X; ?H ) ? V · e
given by
xi(p,q), q 7?? (ei(1,1) ? · · · ? ei(l,1) ) ? · · · ? (ei(1,n) ? · · · ? ei(l,n) ) · e
q=1 p=1
where
i(p, q) = ipq = ?p (q),
(3) For any class
U(gln ) · D(X; ?H ) and
see (2.4), is a bijective
function
is
?
U(gln )-intertwiner.
H , the polynomial D(X; ?)
?l
?l
to e1 ? · · · ? en · e?e by T .
on
mapped
belongs
to
Using the lemma, we have the
Lemma 2.2
It holds that
U(gln ) · D(X; ?) ?
= V · e?e
as a left
U(gln )-module.
In particular,
V · e?e ?
= Vnl (?)
(2.5)
if
?(h) = ??(h) .
V ?
=
M
(2.6)
M?n S ? .
?`nl
Here S ? denotes the irreducible unitary right Snl -module corresponding to ?. We
see that
E
D
?
nl
dim S ? · e = IndS
1
,
S
= K?(ln ) ,
(2.7)
K
K
Snl
where 1K is the trivial representation of K and h?, ?iSnl is the intertwining number
of given representations ? and ? of Snl , and K?µ is the Kostka number. Since
K?(ln ) = 0 unless `(?) 6 n, it follows the
1693
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
Theorem 2.3
The irreducible decomposition
V · e?e ?
=
M
(2.8)
M?n ? S ? · e?e
?`nl
`(?)6n
holds, so that the multiplicity of
M?n
V · e?e
in
is given by
(2.9)
dim S ? · e?e = rankEnd(S ? · e) (e?e).
As a special case, we now obtain the
Theorem 2.4
Let
d = K?(ln ) .
Fix an orthonormal basis
?
{e?j }fj=1
of
S ?,
and denote
?
by {?ij } the matrix coecients relative to this basis. Suppose that the rst d vectors
K
e?1 , . . . , e?d spans S ? . Then the multiplicity of the irreducible representation M?n
(?)
l
in the cyclic module U(gln ) · det (X) is equal to the rank of the matrix
!
?
(?) :=
Fn,l
X
??(h) ?ij? (h)
h?H
?
We refer to the matrix Fn,
l (?) as a
itself does depend on the
rank does not. The trace
(2.10)
.
16i,j6d
transition matrix for
?. The transition matrix
?
choice of the basis {e?j }fj=1 of S ? in
?
of the transition matrix Fn,l
(?) is
?
?
fn,l
(?) = trFn,l
(?) =
X
the theorem, while its
??(h) ? ? (h),
(2.11)
h?H
where ? ? is the
zonal spherical function
? ? (g) =
for ? with respect to K dened by
1 X ?
? (kg) (g ? Snl ).
|K| k?K
(2.12)
?
?
?1 ?
? K
If the matrix Fn,
l (?) is scalar, then Fn, l (?) = d fn, l (?) Id (recall d = dim(S ) )
and hence the multiplicity of M?n in Vnl (?) is completely controlled by the single
?
polynomial fn,l
(?) as in the case where l = 1. Thus it is desirable to obtain a
?
characterization of the triplets (n, l, ?) such that Fn,l
(?) are scalar. The following is
a sucient condition for ? when n and l are given.
Proposition 2.5
NH (K) ?
= Sn ).
NH (K)-module.
Example 2.6
Denote by
NH (K)
The transition matrix
the normalizer of
Fn,? l (?)
is scalar if
K in H ( Notice that
(S ? )K is irreducible as a
(Matsumoto-Wakayama case). If l = 1, then K = 1 and NH (K) = H
so that (S ? )K = S ? is an irreducible NH (K)-module, and hence all the transition
?
?
?
matrices Fn,1
(?) are scalar. In fact, we have Fn,1
(?) = f? (?)I and fn,1
(?) = f ? f? (?).
1694
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
Example 2.7
(hook-type case). If ? = (nl ? r, 1r ) is of hook type (0 6 r 6 n ? 1),
r
then (S
)K ?
= S (n?r,1 ) as NH (K)-modules by [1, Proposition 5.3]. Thus the
(n?r,1r )
transition matrix Fn,l
(?) is scalar. See Appendix for the concrete examples in
this case.
(nl?r,1r )
Example 2.8 (Gelfand pair case). Suppose that (Snl , K) is a Gelfand pair, that
nl
is, the induced representation IndS
K 1K of the trivial representation 1K of K to
Snl is multiplicity-free (see, e.g. [6]). Then (S ? )K is obviously irreducible as an
NH (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) K , we have the
(4,2)
(4,2)
transition matrix F3,2 (?) for M3,2 (?) in V3,2 (?) as
1
(4,2)
F3,2 (?) = (1 + ?)2 diag{2 ? 2? + 3?2 , 1 ? ?, 1 ? ?}.
2
(4,2)
(4,2)
Hence, the multiplicity m3,2 (?) of M3,2 (?) in V3,2 (?) is
?
0 ? = ?1,
?
?
?
?1 ? = 1,
(4,2)
?
m3,2 (?) =
?
2
?
=
(1
±
?5)/3,
?
?
?
3 otherwise.
џ 3. Irreducible decomposition of V2,l (?) and Jacobi
polynomials
When n = 2, as is well known, the pair (S2l , K) is a Gelfand pair, so that the
transition matrices F2,? l (?), where ? ` 2l, `(?) 6 2, are scalar (of size one). If we set
gs = (1, l + 1)(2, l + 2) . . . (s, l + s) ? S2n , then we have
F2,? l (?)
=
trF2,? l (?)
=
X
h?H
?
?(h)
?
? (h) =
l X
l
s=0
s
? ? (gs ) ?s .
Now we write ? = (2l ? r, r) for some r (0 6 r 6 l). The value ? (2l?r,r) (gs ) of the
zonal spherical function is calculated by Bannai and Ito [2, p. 218] as
? (2l?r,r) (gs ) = Qr (s; ?l ? 1, ?l ? 1, l)
?2 r
X
2l ? r + 1
l
s
j r
=
(?1)
,
j
j
j
j
j=0
1695
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
where
N
X
?1 ?1 n ?n ? ? ? ? ? 1 ?? ? 1
N
x
Qn (x; ?, ?, N ) =
(?1)
j
j
j
j
j
j=0
j
Theorem 3.1
Let
l
be a positive integer. It holds that
(2l?r,r)
F2,l
(?)
=
l X
l
s
s=0
=
for
r = 0, 1, . . . , l.
Qr (s; l ? 1, l ? 1, l) ?s
?1
n?l?1
(1 + ?)l?r Pr(?l?1, 2l?2r+1) (1 + 2?)
n
(a,b)
Pn (x) denotes the Jacobi polynomial
n+a
1?x
(a,b)
Pn (x) =
.
2 F1 ?n, a + b + n + 1, a + 1;
2
n
Here
Further, all roots of
(2l?r,r)
F2, l
(?)
are lying on the unit circle
|z| = 1.
Thus we obtain the irreducible decomposition of V2, l (?):
(l, l)
V2, l (?1) ?
= M2 ,
M (2l?r,r)
V2, l (?) ?
M2
, ? 6= ?1,
=
(?l?1, 2l?2r+1)
the sum is taken over r = 0, 1, . . . , l, such that Pr
(1 + 2?) 6= 0.
џ 4. Remarks on related works
For all but nite values of ?, Vnl (?) is equivalent to S l (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 rst point of
view. We introduce the ?-deformation of the quantum determinant in the quantum
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 ? with parameter q as (entries of) transition matrices dened
analogously.
From the second point of view, in [4], we study the case where ? is a reciprocal of
a negative integer. In this case, the ?1/k -determinants satises a ?1/k -analogue
1696
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
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 џ 6 of [4]).
Appendix. Examples of traces of the transition matrices for
hook-types
?
?
(?) of the transition
(?) = trFn,l
Here we give several examples of the trace fn,l
matrices for the case where ? is of hook-type calculated by MAPLE. First remark
?
that we can calculate fn,l
(?) explicitly for ? = (nl), (nl ? 1, 1) as follows:
(nl)
fn,l (?)
n?1
Y
=
(n)
fn,1 (?)l
=
(n)
(n?1,1)
fn,1 (?)l?1 fn,1
(?)
=
(1 + j?)l ,
j=1
(nl?1,1)
fn,l
(?)
l?1
= (n ? 1)(1 ? ?)(1 ? (n ? 1)?)
n?2
Y
(1 + j?)l .
j=1
Here are some other examples:
? (n, l) = (5, 2) :
2
11 2
? ),
2
9
= 4(1 + ?)2 (1 + 2?)(1 + 3?)(1 ? ?)(1 ? 2?)(1 + ? ? ?2 ),
2
2
2
2
= (1 + ?)(1 + 2?) (1 ? ?)(1 ? 2?) (1 ? 6? ).
8,1
f5,2
= 6(1 + ?)2 (1 + 2?)2 (1 + 3?)(1 ? ?)(1 + 2? ?
3
7,1
f5,2
4
6,1
f5,2
? (n, l) = (4, 2) :
2
6,1
f4,2
= 3(1 + ?)2 (1 + 2?)(1 ? ?)(1 + ? ? 4?2 ),
3
5,1
f4,2
= (1 + ?)(1 + 2?)(1 ? ?)(1 ? 2?)(1 ? 3?2 ).
? (n, l) = (3, 2) :
5
4,12
f3,2
= (1 + ?)(1 ? ?)(1 ? ?2 ).
2
1697
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
? (n, l) = (4, 3) :
2
10 2
? ),
3
17
94
= (1 + ?)2 (1 + 2?)2 (1 ? ?)(1 + ? ? 7?2 ? ?3 + ?4 ).
9
9
10,1
= 3(1 + ?)3 (1 + 2?)2 (1 + 3?)(1 ? ?)(1 + ? ?
f4,3
3
9,1
f4,3
? (n, l) = (3, 3) :
2
7,1
f3,3
= (1 + ?)2 (1 + 2?)(1 ? ?)(1 ? 2?2 ).
? (n, l) = (3, 4) :
7
10,12
= (1 + ?)3 (1 + 2?)2 (1 ? ?)(1 ? ?2 ).
f3,4
4
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),
Kinokuniya, Tokyo, 1986, 546568.
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.
4. K. Kimoto and M. Wakayama. Invariant theory for singular ?-determinants,
J. Combin. Theory Ser. A, 2008, vol. 115, No. 1, 131.
5. K. Kimoto and M. Wakayama. Quantum ?-determinant cyclic modules of
Uq (gln ), J. Algebra, 2007, vol. 313, 922956.
6. I. G. Macdonald. Symmetric Functions and Hall Polynomials, 2nd edn., Oxford
University Press, 1995.
7. S. Matsumoto and M. Wakayama. Alpha-determinant cyclic modules of gln (C),
J. Lie Theory, 2006, vol, 16, 393405.
8. T. Shirai and Y. Takahashi. Random point elds associated with certain
Fredholm determinants I: fermion, Poisson and boson point processes, J. Funct.
Anal., 2003, vol. 205, 414463.
9. D. Vere-Jones. A generalization of permanents and determinants, Linear
Algebra Appl., 1988, vol. 111, 119124.
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