On a character formula involving Borel subgroups

On a Character Formula Involving Bore1 Subgroups Mihalis Maliakas 13epartment

of Mathematics

Iiniversity of Athens Panepistimiopolis Athens 15784,

Greece

Let k be a field of characteristic zero, and B the subgroup of GL,(k) of upper triangular matrices. Consider the natural GL,,(k)-module V = k”, and let {el, e2,. . . , e,} be the standard basis of V. We have the flag 0 c v, & v, c ... G v,, where V3 is the subsequence of V spanned by { “1>“2,...> -

ei}. An easy computation

is a B-module.

The (formal)

h,(xl,

x2,...,

shows that each symmetric

character

xj) =

power S,\‘,

of SiVI is

c cl, t ...

xp’x;’

. . . q,,

+a,=i

the ith complete symmetric function on the variables x,, x1, . . . , xj [41. For A = (A,, . . . , A,) E N”, the character of the B-module S,,V, 8 SJ,

@ -1. 8 SAnV,, is the

which

will be denoted

by

product h,{x,)h,Jx,, x2)... h,Jx,, h,. Let k, be the one-dimensional

whose character is ch k, = x:lr,hz ... x,^ll. In this note we give a very elementary equivalent to the main result of Kovacec combinatorics of semistandard tableaux. THEOREM .

direct

proof

and Santana

x2.. . x,,), B-module

of a theorem

[2]; their proof uses

With the above notation we have

ch kA=

c c,TE

(1)

sgn(o) hm(h)~

I-,

where IT,, is the symmetric group on n symbols, sgn(a) denotes the sign of the permutation CT, and a(h) = (A, + a(l) - 1, A, + a(2) - 2 ,..., A, + a(n) - n). (The convention is that h,(,, = 0 is a(A) P N”.) LINEAR

ALGEBRA

AND ITS APPLICATlONS

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274

MIHALIS MALIAKAS

Recently, methods, the teristic yields We begin

Doty [I], Woodcock [5], and the author [3] proved, using various existence of B-resolutions of k, whose Euler-Poincare characthe above character identity. by making two observations:

(a) The right-hand side of (1) is the determinant of the following n X n matrix whose i, j entry is hAi_i+j(xl,. . . , xi): ...

... ...

h Al+n-l(X1) h ,h,+n-2 (bX2)

h&l,:..,

q,>

/

(2) (b) We have the identity

hi(xl,

x2>**.>

xj) - xlhi_l(

xl,

x2,.

. .,

xj) = hi(

for each i E N and j E {1,2,. . . , n], where as usual h,(x,, h_l(x,,..., xj) = 0.

x2,

. . . .

xj)

(3)

. . . , xj) = 1 and

We proceed now by induction on n to show that the determinant of (2) is r+2”2 . . . r,h,* The case n = 1 is trivial. Assuming n > 1, multiply column n - 1 of (2) by x1 and subtract from column n, multiply column n - 2 by x1 and subtract from column n - 1, etc. Using (3), the resulting matrix is I

h*l(xd h**-lb,,

x2)

h A,-n+l (&,...J,,,

...

0 h&2)

h*,-"+P(XP.....

...

4

*.*

0 hA\2+n-2(~2)

h&x,,:..,

xJ (4)

By induction, the determinant of the matrix obtained after deleting the first row and column of (4) is ~2x2 a** x$. Thus the determinant of (4) is +lx> . . . x$, and the proof is complete.

A CHARACTER

275

FORMULA

If A E N” and (Y E Z” with CY= ((Ye,. . . , cc,>, let cA,LI be the number of row semistandard

tableaux of shape A and weight

(Y. Verify that cA,(I equals

the coefficient of the monomial xpl *** x,*n in h,. Thus by considering in (1) rnonomials of weight cx = ((Ye, . . . , a,), one recovers immediately the main result of [2], namely the identity

6A, a where

6, a is the Kronecker

=

~Sgn(~>Co(,qa,

delta.

REFERENCES I

S. R. Doty,

Resolutions

2

A. Kovacec

and A. P. Santana,

of Schur algebra 3

M. Maliakas,

4

I. G. MacDonald,

S(B+),

Resolutions

of B-modules,

In&g.

A combinatorial

Linear

Algebra

and parabolic

Appl.

Math. 5(3):267-283 theorem 197:79-91

Schur algebras,

(1994).

on the Cartan

invariants

(1994). /.

ofAlgebra (to

Symmetric

Functions

and Hull Polynomials,

Bore1 Schur

algebras,

Comm.

Clarendon,

appear). Oxford,

1979. 5

D. J. Woodcock,

Algebra

22:1703-1721

Received 15 September 1994; final manuscript accepted 2 February 1995

(1994).