Representation Theory of Reductive Normal Algebraic Monoids

Doty, Stephen

doi:10.1090/s0002-9947-99-02462-9

Cited by 9 publications

(3 citation statements)

References 21 publications

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“…Although Doty alluded to the fact that Frobenius Reciprocity holds for monoids [3], he left it without proof. For completeness, we give an explicit proof.…”

Section: Frobenius Reciprocitymentioning

confidence: 99%

Convolution Algebras for Finite Reductive Monoids

Jared¹,

McDonald²,

O’Brien³

et al. 2018

Preprint

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For an arbitrary finite monoid M and subgroup K of the unit group of M, we prove that there is a bijection between irreducible representations of M with nontrivial K-fixed space and irreducible representations of H K , the convolution algebra of K × K-invariant functions from M to F, where F is a field of characteristic not dividing |K|. When M is reductive and K = B is a Borel subgroup of the group of units, this indirectly provides a connection between irreducible representations of M and those of F[R], where R is the Renner monoid of M. We conclude with a quick proof of Frobenius Reciprocity for monoids for reference in future papers.

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“…Although Doty alluded to the fact that Frobenius Reciprocity holds for monoids [3], he left it without proof. For completeness, we give an explicit proof.…”

Section: Frobenius Reciprocitymentioning

confidence: 99%

Convolution Algebras for Finite Reductive Monoids

Jared¹,

McDonald²,

O’Brien³

et al. 2018

Preprint

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“…Indeed, since then highest weight categories have attracted large interest in representation theory (see e.g. [Dot99,Gua03,BE09,BT18,Cou20]). Another very interesting result in [CPS88] was the Brauer-Humphreys reciprocity theorem [CPS88,Thm.…”

Section: Introductionmentioning

confidence: 99%

Stratifying systems and Jordan-Hölder extriangulated categories

Brüstle¹,

Hassoun²,

Shah³

et al. 2022

Preprint

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Stratifying systems, which have been defined for module, triangulated and exact categories previously, were developed to produce examples of standardly stratified algebras. A stratifying system Φ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory F(Φ) of objects admitting a composition series-like filtration with factors in Φ has the Jordan-Hölder property on these filtrations.This article has two main aims. First, we introduce notions of subobjects, simple objects and composition series relative to the extriangulated structure in order to define a Jordan-Hölder extriangulated category. Moreover, we characterise these categories in terms of the associated Grothendieck monoid and Grothendieck group.Second, we develop a theory of stratifying systems in extriangulated categories. We define projective stratifying systems and show that every stratifying system Φ in an extriangulated category is part of a projective one (Φ, Q). We prove that F(Φ) is a Jordan-Hölder extriangulated category when (Φ, Q) satisfies some extra conditions.

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“…where B is the Zariski closure of B in M; we will call B the Borel submonoid determined by B. Although its combinatorics and geometry are relatively less explored compared to that of the ambient reductive monoid, the Borel submonoid is a very important object for the study of the representation theory of M [11,Theorem 3.4]. In the special case of the linear algebraic monoid of n × n matrices, the B × B-orbits in B are parametrized by the set partitions of {1, .…”

Section: Introductionmentioning

confidence: 99%

On the Borel Submonoid of a Symplectic Monoid

Can¹,

Houser²,

Wolfe³

2020

Preprint

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In this article, we study the Bruhat-Chevalley-Renner order on the complex symplectic monoid M Sp n . After showing that this order is completely determined by the Bruhat-Chevalley-Renner order on the linear algebraic monoid of n × n matrices M n , we focus on the Borel submonoid of M Sp n . By using this submonoid, we introduce a new set of type B set partitions. We determine their count by using the "folding" and "unfolding" operators that we introduce. We show that the Borel submonoid of a rationally smooth reductive monoid with zero is rationally smooth. Finally, we analyze the nilpotent subsemigroups of the Borel semigroups of M n and M Sp n . We show that, contrary to the case of M Sp n , the nilpotent subsemigroup of the Borel submonoid of M n is irreducible.

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Representation Theory of Reductive Normal Algebraic Monoids

Cited by 9 publications

References 21 publications

Convolution Algebras for Finite Reductive Monoids

Convolution Algebras for Finite Reductive Monoids

Stratifying systems and Jordan-Hölder extriangulated categories

On the Borel Submonoid of a Symplectic Monoid

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