Schubert calculus and torsion explosion

Williamson, Geordie

doi:10.1090/jams/868

Cited by 104 publications

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“…By a careful analysis of the combinatorics of moment graph sheaves Fiebig was able to give an explicit (enormous) bound above which Lusztig's conjecture holds [Fie12] and establish the multiplicity one case [Fie10b]. [Wil16c,Wil16b]. The upshot is that the above bounds (like p ě h or p ě 2h´3) are much too optimistic.…”

Section: Status Of Lusztig's Character Formula We Give a Brief Summamentioning

confidence: 99%

“…The upshot is that the above bounds (like p ě h or p ě 2h´3) are much too optimistic. In fact, recent advances in number theory imply that there is no polynomial bound in the Coxeter number for the validity of Lusztig's conjecture (see the appendix to [Wil16c] by Kontorovich, McNamara and the author). We will discuss these results in more detail in the next section.…”

Section: Status Of Lusztig's Character Formula We Give a Brief Summamentioning

confidence: 99%

“…Counter-examples to expected bounds. In this section we describe the results of [Wil16c,Wil16b] in the context of the Finkelberg-Mirković conjecture.…”

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confidence: 99%

“…The main idea of [Wil16c] is that certain structure constants occurring in Schubert calculus for the cohomology ring H˚pX _ ; Zq also occur as torsion in local integral intersection cohomology in much higher rank groups. Using these ideas it is shown, for example, that any prime number dividing any entry of a word of length ℓ in the semi-group Bˆ1 1 0 1˙,ˆ1 0 1 1˙F Ă SL 2 pZq occurs as torsion in the stalks or costalks of some IC Z x,B _ on the flag variety of GL 5`3ℓ pCq.…”

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confidence: 99%

“…The main result of [Wil16c] uses a formula for certain entries of intersection forms obtained by the author and He [HW15]. This result uses the theory of generators and relations for Soergel bimodules [EW13] in a crucial way.…”

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confidence: 99%

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Algebraic representations and constructible sheaves

Williamson

2017

Jpn. J. Math.

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Abstract. These are notes for my Takagi lecture at the University of Tokyo in November, 2016. I survey what is known about simple modules for reductive algebraic groups. The emphasis is on characteristic p ą 0 and Lusztig's character formula. I explain ideas connecting representations and constructible sheaves (Finkelberg-Mirković conjecture) in the spirit of the Kazhdan-Lusztig conjecture. I also discuss a conjecture with S. Riche (a theorem for GLn) which should eventually make computations more feasible.

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Section: Status Of Lusztig's Character Formula We Give a Brief Summamentioning

confidence: 99%

Section: Status Of Lusztig's Character Formula We Give a Brief Summamentioning

confidence: 99%

“…Counter-examples to expected bounds. In this section we describe the results of [Wil16c,Wil16b] in the context of the Finkelberg-Mirković conjecture.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Algebraic representations and constructible sheaves

Williamson

2017

Jpn. J. Math.

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Some Tables of Right Set Properties in Affine Weyl Groups of Type A

Scott

Zell

2019

Springer Proceedings in Mathematics &Amp; Statistics

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The tables of the title are a first attempt to understand empirically the sizes of certain distinguished sets, introduced by Hankyung Ko, of elements in affine Weyl groups. The distinguished sets themselves each have a largest element w, and all other elements are constructible combinatorially from that largest element. The combinatorics are given in the language of right sets, in the sense of Kazhdan-Lusztig. Collectively, the elements in a given distinguished set parameterize highest weights of possible modular composition factors of the "reduction modulo p" of a p th root of unity irreducible characteristic 0 quantum group module. Here p is a prime, subject to conditions discussed below, in some cases known to be quite mild. Thus, the sizes of the distinguished sets in question are relevant to estimating how much time might be saved in any future direct approach to computing irreducible modular characters of algebraic groups from larger irreducible characters of quantum groups. Actually, Ko has described two methods for obtaining potentially effective systems of such sets. She has proved one method to work at least for all primes p as large as the Coxeter number h, in a context she indicates largely generalizes to smaller p. The other method, which produces smaller distinguished sets, is known for primes p ≥ h for which the Lusztig character formula holds, but is currently unknown to be valid without the latter condition. In the tables of this paper we calculate, for all w indexing a (p-)regular highest weight in the (p-)restricted parallelotope, distinguished set sizes for both methods, for affine types A 3 , A 4 , and A 5 . To keep the printed version of this paper sufficiently small, we only use those w indexing actual restricted weights in the A 5 case. The sizes corresponding to the two methods of Ko are listed in columns (6) and (5), respectively, of the tables. We also make calculations in column (7) for a third, more "obvious" system of distinguished sets (see part (1) of Proposition 1 below), to indicate how much of an improvement each of the first two systems provides. Finally, all calculations have been recently completed for affine type A 6 , and the restricted cases listed in this paper as a final table.

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Representations of Symmetric Groups

Craven

2019

Universitext

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Schubert calculus and torsion explosion

Cited by 104 publications

References 50 publications

Algebraic representations and constructible sheaves

Algebraic representations and constructible sheaves

Some Tables of Right Set Properties in Affine Weyl Groups of Type A

Representations of Symmetric Groups

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