On refinements of the Bruhat decomposition

Simion, Iulian I.

doi:10.1007/s10801-015-0626-x

Cited by 2 publications

(4 citation statements)

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“…For concretely determining these intersections in both the U xU -form and zU y −1 U -form which are required for our purposes, one needs to translate the inductive descriptions into the two forms. By making use of the D j described in [Sim16], we illustrate how the algorithm works in general. Such a procedure has been implemented in Python [Chevalley].…”

Section: U -Double Cosetsmentioning

confidence: 99%

“…From results of Curtis [Cur88,Cur09,Cur15] which rely on work of Deodhar [Deo85], we can decompose the intersection into disjoint sets indexed by distinguished subexpressions of the fixed reduced expression of x. Building on results of [Sim16], and exploiting the theory of the invariants for distinguished subexpressions, we first determine the intersections for certain t w and then we extend this result to all elements of N G (T ). This decomposition method resembles the principle of "collection from the left", see for example [Sim14,§9.4].…”

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confidence: 99%

“…An algorithm for obtaining these constants was provided by C. Curtis (see [Cur15]). A variation of this algorithm, with different choices for U -coset representatives was described in [Sim16]. We use this latter version (see Section 2) to obtain the elements ( 5)…”

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confidence: 99%

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On Refined Bruhat Decompositions and Endomorphism Algebras of Gelfand-Graev Representations

Paolini

Simion

2019

Algebr Represent Theor

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Let G be a finite reductive group defined over Fq, with q a power of a prime p. Motivated by a problem recently posed by C. Curtis, we first develop an algorithm to express each element of G into a canonical form in terms of a refinement of a Bruhat decomposition, and we then use the output of the algorithm to explicitly determine the structure constants of the endomorphism algebra of a Gelfand-Graev representation of G when G = PGL3(q) for an arbitrary prime p, and when G = SO5(q) for p odd.Let G be the fixed point subgroupḠ F of a reductive algebraic groupḠ under a Frobenius endomorphism F . A special role in the representation theory of finite reductive groups is played by Gelfand-Graev representations. These are certain representations induced from a linear character of a maximal unipotent subgroup of G. If the center of the ambient algebraic groupḠ is connected, then a Gelfand-Graev representation Γ is unique up to conjugacy and the irreducible components of Γ were described in [DL76]. In general, these representations were parametrized and decomposed in [DM91, Section 14] and [DLM92, Section 3].Denote by H the Hecke algebra (that is, the G-endomorphism algebra) of the module affording Γ. Since Γ is multiplicity free [Car85, Theorem 8.1.3], the algebra H is abelian. C. Curtis parametrized in [Cur93] the irreducible representations of H by pairs (T, θ) where T =T F for an F -stable maximal torusT ofḠ and where θ is an irreducible character of T . Each irreducible representation f T,θ of the algebra H is shown to have the following factorization [Cur93, Theorem 4.2], f T,θ =θ • f T , where f T : H →Q ℓ T is a homomorphism of algebras andθ is the linear extension of θ to the group algebraQ ℓ T . The homomorphism f T is independent of θ. Curtis' homomorphism f T was considered in [BK08] in the context of ℓ-modular Gelfand-Graev representations with ℓ = p.Here it is shown that if ℓ does not divide the order of the Weyl group of G, then the behavior of such endomorphism algebras is generic along all prime powers q.In order to obtain a constructive description of each f T , one needs to consider structure constants of H with respect to some basis. Building on work of Kawanaka [Kaw77] and Deodhar [Deo85], Curtis described in [Cur15] an algorithm for obtaining structure constants for H with respect to a standard basis parametrized by certain elements in N G (T ). In general, the determination of the structure

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Section: U -Double Cosetsmentioning

confidence: 99%

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confidence: 99%

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On Refined Bruhat Decompositions and Endomorphism Algebras of Gelfand-Graev Representations

Paolini

Simion

2019

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“…Formulas for the structure constants based on different algorithms and a different set of representatives of the cosets of U were obtained by Simion [2015].…”

Section: Introductionmentioning

confidence: 99%

Notes on the structure constants of Hecke algebras of induced representations of finite Chevalley groups

Curtis

2015

Pacific J. Math.

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On refinements of the Bruhat decomposition

Cited by 2 publications

References 9 publications

On Refined Bruhat Decompositions and Endomorphism Algebras of Gelfand-Graev Representations

On Refined Bruhat Decompositions and Endomorphism Algebras of Gelfand-Graev Representations

Notes on the structure constants of Hecke algebras of induced representations of finite Chevalley groups

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