Green Vertex Theory, Green Correspondence, and Harish-Chandra Induction

Gruber, Jochen

doi:10.1006/jabr.1996.0384

Cited by 6 publications

(3 citation statements)

References 3 publications

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“…We obtainĀ ′ = {0, 1, 3,13,14,15,19,20, 21}, again using n = 2. Next, (Ā ′ ) 1 = {0, 1, 6, 7, 9, 10} and (Ā) 2 = {0, 7, 10}, wich are β-sets for (5 2 , 4 2 ) and (8,6) respectively.…”

Section: 2mentioning

confidence: 99%

Harish-Chandra Series in Finite Unitary Groups and Crystal Graphs

Gerber

Hiß

Jacon³

2015

International Mathematics Research Notices

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Abstract. The distribution of the unipotent modules (in nondefining prime characteristic) of the finite unitary groups into Harish-Chandra series is investigated. We formulate a series of conjectures relating this distribution with the crystal graph of an integrable module for a certain quantum group. Evidence for our conjectures is presented, as well as proofs for some of their consequences for the crystal graphs involved. In the course of our work we also generalize Harish-Chandra theory for some of the finite classical groups, and we introduce their Harish-Chandra branching graphs.

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Section: 2mentioning

confidence: 99%

Harish-Chandra Series in Finite Unitary Groups and Crystal Graphs

Gerber

Hiß

Jacon³

2015

International Mathematics Research Notices

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“…Some general results on identifying Harish-Chandra series of some unipotent modules using the formalism of Hom functors and q-Schur algebras were proved in [9] but the particular statement we prove next seems to be new. A special case, when e = 3 and λ = 2 3 , was shown in [29, Section 2], where it is deduced from [24,Lemma 3.16] and [33,Proposition 2.3.5].…”

Section: 2mentioning

confidence: 99%

On modular Harish-Chandra series of finite unitary groups

Norton¹

2019

Preprint

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In the modular representation theory of finite unitary groups when the characteristic ℓ of the ground field is a unitary prime, the sl e -crystal on level 2 Fock spaces graphically describes the Harish-Chandra branching of unipotent representations restricted to the tower of unitary groups. However, how to determine the cuspidal support of an arbitrary unipotent representation has remained an open question. We show that for ℓ sufficiently large, the sl ∞ -crystal on the same level 2 Fock spaces provides the remaining piece of the puzzle for the full Harish-Chandra branching rule.

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“…Such an embedding was explicitly described in [28], 2.2. Such an embedding was explicitly described in [28], 2.2.…”

Section: 2mentioning

confidence: 99%

Decomposition numbers of finite classical groups for linear primes.

1997

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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A finite classical group is a unitary, conformal symplectic or special orthogonal group G, defined over some finite field with q elements (respectively q 2 , if G is unitary). An odd prime t not dividing q is called linear for G, if the order of q modulo t is odd. It is shown that, for linear primes /, the /-decomposition numbers of G can be computed from those of general linear groups.As a corollary we obtain a canonical labeling of the irreducible Brauer characters of G by a certain distinguished subset of ordinary characters of G. Moreover, the decomposition numbers are bounded above by the order of the Weyl group of G.To obtain the results we generalize the #-Schur algebras introduced by Dipper and James to other types of Iwahori-Hecke algebras. Over suitable ground rings, the new q-Schur algebras turn out to be Morita equivalent to direct sums of tensor products of #-Schur algebras of type A.*) This paper is a contribution to the DFG project "Algorithmic number theory and algebra". The authors gratefully acknowledge financial support from the DFG. Most of the results in this paper have been announced in [23] and [31]. The resultsfor the unipotent characters were obtained by the first author in bis Dissertation [27].Let us now comment on the principal ideas behind our work. An irreducible Brauer character of G (in non-describing characteristic) is cuspidal if and only if the corresponding projective indecomposable module is not a direct summand of a projective module, Harish-Chandra induced from a proper Levi subgroup. Thus in order to obtain a classification of the cuspidal Brauer characters, one is lead to induce projective indecomposable modules from proper Levi subgroups in a systematic way. Actually, we do not induce projective modules but certain factors of these, called /-regulär quotients (Sections 3 and 6). These /-regulär quotients are easier to control than the projectives themselves.

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Green Vertex Theory, Green Correspondence, and Harish-Chandra Induction

Cited by 6 publications

References 3 publications

Harish-Chandra Series in Finite Unitary Groups and Crystal Graphs

Harish-Chandra Series in Finite Unitary Groups and Crystal Graphs

On modular Harish-Chandra series of finite unitary groups

Decomposition numbers of finite classical groups for linear primes.

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