Generalized Harish-Chandra theory

Broué, Michel; Malle, Gunter

doi:10.1017/cbo9780511600623.006

Cited by 64 publications

(146 citation statements)

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“…If b = NE(a), then N(a) is attached to neither its NE nor its SE neighbour, so is the end node of its tile, and in particular dp(N(a)) = 0. On the other hand dp(a) > dp(b) 0, contradicting (2). If instead b = SE(a), then NE(a) is attached to neither its NW nor its SW neighbour, so is the start node of its tile, and has depth 0; but dp(b) > dp(a) 0, and again (2) is contradicted.…”

Section: Proofmentioning

confidence: 91%

“…If both SE(l) and SW(l) lie in μ, then l is an addable node and we are in case (1). If neither lies in μ, then S(l) is a removable node, and we are in case (2). If SE(l) ∈ μ / SW(l), then we are in case (3), while if SE(l) / ∈ μ SW(l) then we are in case (4).…”

Section: Proof Of Theorem 52mentioning

confidence: 99%

“…The cover-inclusive property follows easily from the corresponding property of T , and in particular the fact [Theorem 3.1 (2)] that depth weakly decreases down columns in a cover-inclusive tiling. Moreover, φ 3 (T ) has no big tile starting or ending in column j, so lies in the codomain.…”

Section: Proposition 38mentioning

confidence: 99%

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Dyck tilings and the homogeneous Garnir relations for graded Specht modules

Fayers

2016

J Algebr Comb

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Suppose λ and μ are integer partitions with λ ⊇ μ. Kenyon and Wilson have introduced the notion of a cover-inclusive Dyck tiling of the skew Young diagram λ\μ, which has applications in the study of double-dimer models. We examine these tilings in more detail, giving various equivalent conditions and then proving a recurrence which we use to show that the entries of the transition matrix between two bases for a certain permutation module for the symmetric group are given by counting coverinclusive Dyck tilings. We go on to consider the inverse of this matrix, showing that its entries are determined by what we call cover-expansive Dyck tilings. The fact that these two matrices are mutual inverses allows us to recover the main result of Kenyon and Wilson. We then discuss the connections with recent results of Kim et al., who give a simple expression for the sum, over all μ, of the number of cover-inclusive Dyck tilings of λ\μ. Our results provide a new proof of this result. Finally, we show how to use our results to obtain simpler expressions for the homogeneous Garnir relations for the universal Specht modules introduced by Kleshchev, Mathas and Ram for the cyclotomic quiver Hecke algebras.

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Section: Proofmentioning

confidence: 91%

Section: Proof Of Theorem 52mentioning

confidence: 99%

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Dyck tilings and the homogeneous Garnir relations for graded Specht modules

Fayers

2016

J Algebr Comb

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“…Let Φ = T 1 T 2 T 3 . It follows from the computation of the generic degrees and the character table of H q (w) in [BMa,5.A] that a representation is special if (and only if) the elements {(T i ) i=1,2,3 , (T i T j ) i=1,2,3, j=1+(i mod 3) , (Φ i ) i=1,2,3 } have trace zero on it. The quadratic relations show that B = {(T i ) i=1,2,3 , (T i Φ) i=1,2,3 , (Φ i ) i=1,2,3 } generates the same subspace of H q (w) as these elements.…”

Section: All S In the Support Of T Divide V On The Left And That In mentioning

confidence: 99%

“…It remains to see that the virtual representation i (−1) i H i c (X(c)) of H(c) is special. But, by e.g., [BMa,2.2], the symmetrizing trace on H(c) is characterized by its vanishing on T i for i = 1, . .…”

Section: §1 Introductionmentioning

confidence: 99%

Endomorphisms of Deligne-Lusztig Varieties

Digne¹,

Michel²

2006

Nagoya Mathematical Journal

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Abstract. We study some conjectures on the endomorphism algebras of the cohomology of Deligne-Lusztig varieties which are a refinement of those of [BMi]. §1. Introduction Let G be a connected reductive algebraic group, defined over an algebraic closure F of a finite field of characteristic p. Let F be an isogeny on G such that some power F δ is the Frobenius endomorphism attached to a split F q δ -structure on G (where q is a real number such that q δ is a power of p). The finite group G F of fixed points under F is called a finite group of Lie type. When considering a simple group which is not a Ree or Suzuki group we may take F to be already a Frobenius endomorphism.Let W be the Weyl group of G and let B (resp. B + ) be the corresponding braid group (resp. monoid). The canonical morphism of monoids β : B + → W has a section that we denote by w → w: it sends an element of W to the only positive braid w such that β(w) = w and such that the length of w in B + is the same as the Coxeter length of w; we write W = {w | w ∈ W } and S = {s | s ∈ S} where S is a set of Coxeter generators for W .Let us recall how in [BMi] a "Deligne-Lusztig variety" is attached to each element of B + . Let B be the variety of Borel subgroups of G. The orbits of B×B are in natural bijection with W . Let O(w) be the orbit corresponding to w ∈ W . Let b ∈ B + and let b = w 1 · · · w n be a decomposition of b as a product of elements of W. To such a decomposition we attach the variety {(B 1 , . . . , B n+1 ) | (B i , B i+1 ) ∈ O(w i ) and B n+1 = F (B 1 )}. It is shown in [De, p. 163] that the varieties attached to two such decompositions of b are canonically isomorphic. The projective limit of this system

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