The character formula of irreducible representations of gl(2/1)

THE CHARACTER FORMULA OF IRREDUCIBLE REPRESENTATIONS OF gl(2|1)

Nguyen Thi Phuong Dung

Banking Academy

Tãm t¾t

Cho V lµ siªu kh«ng gian víi siªu chiÒu (m|n). Khi ®ã, ta cã siªu nhãm tuyÕn tÝnh tæng qu¸t GL(m|n).

Trong tr­êng hîp m = 0 hoÆc n = 0, c¸c biÓu diÔn bÊt kh¶ qui cña siªu nhãm nµy ®· hoµn toµn ®­îc x©y

dùng. Theo Schur and Weyl, ta cã c«ng thøc tÝnh ®Æc tr­ng cho c¸c biÔu diÔn bÊt kh¶ qui nh­ lµ ®Þnh thøc cña

c¸c tensor ®èi xøng Si

. Trong tr­êng hîp c¶ m vµ n ®Òu kh¸c 0, ®iÒu nµy ch­a ®­îc chøng minh. Trong bµi

b¸o nµy chóng t«i ®­a ra ®­îc c«ng thøc t­¬ng tù cho tr­êng hîp m = 2 vµ n = 1.

Tõ kho¸: Nhãm tuyÕn tÝnh, c«ng thøc ®Æc trung, Verma module, biÓu diÔn ®iÓn h×nh, biÓu diÔn kh«ng

®iÓn h×nh.

1 Introduction

Let V be a super vector space over a field k of characteristic of 0. The super group GL(V ) of linear

automorphisms of V is the subgroup of the semi-group End(V ) of endomorphisms with invertible super￾determinant. In [12] Manin introduced the following Koszul complex K to define the super determinat.

Its (k, l)-term is given by Kk,l := Λk ⊗S

∗

l

, where Λn and Sn are the n-th homogeneous components of

the exterior and the symmetric tensor algebra on V . The differential dk,l : Kk,l −→ Kk+1,l+1 is given

by

dk,l(h ⊗ ϕ) = X

i

h ∧ xi ⊗ ξ

i

· ϕ

where Xl

, Yk are the symetrizer and anti-symmetrizer operators.

In the case n = 0, m 6= 0, irreducible representations of G have classified and indexed by partitions

(λ1, λ2, · · · , λm) : λi ≥ λi+1, λi ∈ Z. Particular, the character of all irreducible representations of G

are given by the determinant of Si

.

In [15], by using the Koszul complexes, we constructed all irreducible representations of GL(2|1).

In [9], Kac proved that any finite dimensional irreducible representations of Lie super algebra gl(V ) is a

Verma module. He divided irreducible representations of gl(V ) into two classes, typical representations

and atypical representations. By using Verma module, Kac gave explicit construction of all typical

representation of gl(V ). A character formula for all typical representation was also obtained. In [15] Su

and Zhang gave a character formula for all finite-dimensional irreducible representations of gl(V ). The

formula character of irreducible representations by using determinant of Si

is however not known.

The aim of this work is to give a formulla character using by determinant of all irreducible representations

in case the super-dimension of V is (2|1).

1

*Tel: 0976605305, e-mail: [email protected]

151Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn