Soluble groups of matrices over an arbitrary field

Short Generating Functions for some Semigroup Algebras

The Electronic Journal of Combinatorics ◽

10.37236/1729 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 23

Author(s):

Graham Denham

Keyword(s):

Rational Function ◽

Generating Functions ◽

Hilbert Series ◽

Simple Expression ◽

Arbitrary Field ◽

Graded Algebra ◽

Semigroup Algebras ◽

Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

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Computational approaches to the theory of finite soluble groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700027386 ◽

1988 ◽

Vol 38 (1) ◽

pp. 157-158 ◽

Cited By ~ 1

Author(s):

Stephen Peter Glasby

Keyword(s):

Soluble Groups ◽

Computational Approaches

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Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras

Open Mathematics ◽

10.1515/math-2019-0117 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1381-1391

Author(s):

Keli Zheng ◽

Yongzheng Zhang

Keyword(s):

Lie Superalgebras ◽

Arbitrary Field ◽

Special Linear ◽

Highest Weight ◽

Irreducible Modules

Abstract Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.

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