This theorem has numerous applications to representation theory and other areas of mathematics (cf. [2, 3, 4, 5, 13]). The proof originally given by Steinberg in [15] involved the algebra of holomorphic functions on V and a subtle characterisation of reflection groups. Another proof has been given by Serre [1, Chapter V, Example 8, page 139], which is no more elementary, and which is algebraic ...

The h-deformed quantum plane is a counterpart of the q-deformed one in the set of quantum planes which are covariant under those quantum deformations of GL(2) which admit a central determinant. We have investigated the noncommutative geometry of the h-deformed quantum plane. There is a 2-parameter family of torsion-free linear connections, a 1-parameter sub-family of which are compatible with a...

We generalize the generating formula for plane partitions known as MacMahon’s formula as well as its analog for strict plane partitions. We give a 2-parameter generalization of these formulas related to Macdonald’s symmetric functions. The formula is especially simple in the Hall-Littlewood case. We also give a bijective proof of the analog of MacMahon’s formula for strict plane partitions.

This paper describes a technique for analyzing shapes with potential symmetry employing symmetric subsets. The proposed technique assumes that the shapes interested have potential bilateral symmetry. The notion of potential symmetry is given to those shapes with approximate or original symmetry. As respective examples, we can imagine a human face and a partial broken vase. Although their symmet...

In this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric Connes amenability. We determine symmetric module amenability and symmetric Connes amenability of some concrete Banach algebras. Indeed, it is shown that $ell^1(S)$ is  a symmetric $ell^1(E)$-module amenable if and only if $S$ is amenable, where $S$ is an inverse semigroup with subsemigr...