# CANONICAL LEFT CELLS AND THE SHORTEST LENGTH ELEMENTS IN THE DOUBLE COSETS OF WEYL GROUPS

• Kwon, Nam-Hee (Department of Mathematics, Daegu University)
• Accepted : 2011.02.08
• Published : 2011.03.25
• 63 1

#### Abstract

Let G be the general linear group GL(n,$\mathbb{C}$), $W_0$ the Weyl group of G and W the extended a neWeyl group of G. Then it is well-known that W is a union of the double cosets $W_{0x}W_0$ as x moves over the set of dominant weights of W. It is also known that each double coset $W_{0x}W_0$ contains a unique element $m_x$ of the shortest length. These shortest length elements belong to what are called the canonical left cells. However, it is still an open problem to find the canonical left cell containing a given $m_x$. One of the mai purposes of this paper is to introduce a new approach to attack this question. In particular, we will present a conjecture which explicitly describes the canonical left cells containing an element $m_x$. We will show that our conjecture is true for some specific types of $m_x$.

#### Keywords

Weyl group;two-sided cell;canonical left cell

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