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GRASSMANNIAN SEMIGROUPS AND THEIR REPRESENTATIONS

Part of: Basic linear algebra Representation theory of rings and algebras Semigroups Special varieties Algebraic combinatorics

Published online by Cambridge University Press:  27 March 2018

VICTOR CAMILLO  and
MIODRAG C. IOVANOV
VICTOR CAMILLO*
Affiliation:
University of Iowa, MacLean Hall, Iowa City, IA 52246, USA email [email protected]
MIODRAG C. IOVANOV
Affiliation:
University of Iowa, MacLean Hall, Iowa City, IA 52246, USA email [email protected], [email protected]
*
✉[email protected]
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Abstract

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The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the $\text{Gl}_{n}(\mathbb{K})$ action on the $n\times n$ matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one–one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in $M_{n}(\mathbb{K})$. We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams.

Keywords

semigroupGrassmanniansimultaneous formechelon formrepresentations

MSC classification

Primary: 15A30: Algebraic systems of matrices 20M20: Semigroups of transformations, etc. 14M15: Grassmannians, Schubert varieties, flag manifolds
Secondary: 16G99: None of the above, but in this section 05E99: None of the above, but in this section 20M30: Representation of semigroups; actions of semigroups on sets
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 104 , Issue 3 , June 2018 , pp. 308 - 337
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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