A generalization of Fulton’s conjecture for arbitrary groups

Belkale, Prakash; Ressayre, Nicolas

doi:10.1007/s00208-011-0728-2

Cited by 15 publications

(35 citation statements)

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“…We will see in Corollary 2.3 that this is an equality of divisors. It is easy to see that r Dpj, vq is irreducible by the same argument as for the irreducibility of X (see e.g., [BKR12]).…”

Section: Note That Rmentioning

confidence: 95%

“…Finally we recall the generalization of Fulton's conjecture proved in [BKR12] in two equivalent forms:…”

Section: 5mentioning

confidence: 99%

“…(For the scheme structures, see [BKR12].) As shown in [BKR12], the projection map π 1 : X 1 Ñ ś s i"1 pG{Q 1 w i q is birational. Moreover, because the X w are all normal, each X w i zZ w i has codimension ě 2 in X w i ; accordingly, X 1 zZ 1 has codimension ě 2 in X 1 .…”

Section: Enlargement Of Schubert Cells Letmentioning

confidence: 99%

“…In particular, the ramification results of [BKR12] play a crucial role in the construction of the induction operation (this phenomenon does not appear for type A maximal parabolics). The generalized Fulton conjecture proved in [BKR12] plays a key role in the proof of Theorem 1.6. We build upon earlier work on the minimal set of inequalities defining Γps, Kq [Bel01, BK06,Res10], and the attendant ramification theory in enumerative problems [Bel04, BK06,BKR12].…”

Section: Introductionmentioning

confidence: 99%

“…We build upon earlier work on the minimal set of inequalities defining Γps, Kq [Bel01, BK06,Res10], and the attendant ramification theory in enumerative problems [Bel04, BK06,BKR12]. The induction operation is inspired by [Res10], the basic divisors Dpj, vq ((1.6) below), which are shown here to give extremal rays, appear implicitly in [BKR12]. These basic divisors can also be seen to correspond to E j considered in [Res10,Section 4.1] for an optimal choice of X o (as in loc.…”

Section: Introductionmentioning

confidence: 99%

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Extremal Rays in the Hermitian Eigenvalue Problem for Arbitrary Types

Belkale

Kiers

2019

Transformation Groups

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The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of G " SLpnq . There is a polyhedral cone (the "eigencone") determining the possible answers to the problem. These eigencones can be defined for arbitrary semisimple groups G, and also control the (suitably stabilized) problem of existence of non-zero invariants in tensor products of irreducible representations of G.We give a description of the extremal rays of the eigencones for arbitrary semisimple groups G by first observing that extremal rays lie on regular facets, and then classifying extremal rays on an arbitrary regular face. Explicit formulas are given for some extremal rays, which have an explicit geometric meaning as cycle classes of interesting loci, on an arbitrary regular face, and the remaining extremal rays on that face are understood by a geometric process we introduce, and explicate numerically, called induction from Levi subgroups. Several numerical examples are given. The main results, and methods, of this paper generalize [Bel17] which handled the case of G " SLpnq.

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“…We will see in Corollary 2.3 that this is an equality of divisors. It is easy to see that r Dpj, vq is irreducible by the same argument as for the irreducibility of X (see e.g., [BKR12]).…”

Section: Note That Rmentioning

confidence: 95%

“…Finally we recall the generalization of Fulton's conjecture proved in [BKR12] in two equivalent forms:…”

Section: 5mentioning

confidence: 99%