Irreducible representations of Hecke–Kiselman monoids

Let K[HKΘ] denote the Hecke–Kiselman algebra of a finite oriented graph Θ over an algebraically closed field K. All irreducible representations, and the corresponding maximal ideals of K[HKΘ], are characterized in case this algebra satisfies a polynomial identity. The latter condition corresponds to...

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Bibliographic Details
Published in:Linear algebra and its applications 2022-05, Vol.640, p.12-33
Main Author: Wiertel, Magdalena
Format: Article
Language:English
Subjects:
Algebra of matrix type
Fields (mathematics)
Group theory
Hecke–Kiselman algebra
Irreducible representations
Linear algebra
Matrix algebra
Monoid
Monoids
Polynomials
Prime ideals
Representations
Semigroups
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Summary:Let K[HKΘ] denote the Hecke–Kiselman algebra of a finite oriented graph Θ over an algebraically closed field K. All irreducible representations, and the corresponding maximal ideals of K[HKΘ], are characterized in case this algebra satisfies a polynomial identity. The latter condition corresponds to a simple condition that can be expressed in terms of the graph Θ. The result shows a surprising similarity to the classical results on representations of finite semigroups; namely every representation either comes form an idempotent in the Hecke–Kiselman monoid HKΘ (and hence it is 1-dimensional), or it comes from certain semigroup of matrix type. The case when Θ is an oriented cycle plays a crucial role; the prime spectrum of K[HKΘ] is completely characterized in this case.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2022.01.007