Featured Researches

Representation Theory

An introduction to the lattice of torsion classes

In this expository note, I present some of the key features of the lattice of torsion classes of a finite-dimensional algebra, focussing in particular on its complete semidistributivity and consequences thereof. This is intended to serve as an introduction to recent work by Barnard-Carroll-Zhu and Demonet-Iyama-Reading-Reiten-Thomas.

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Representation Theory

Andrews-Gordon type series for the level 5 and 7 standard modules of the affine Lie algebra A (2) 2

We give Andrews-Gordon type series for the principal characters of the level 5 and 7 standard modules of the affine Lie algebra A (2) 2 . We also give conjectural series for some level 2 modules of A (2) 13 .

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Representation Theory

Annihilators and associated varieties of Harish-Chandra modules for Sp(p,q)

We show how to compute annihilators and associated varieties of simple Harish-Chandra modules for Sp(p,q) with trivial infinitesimal character from a pair of domino tableaux attached to a parameter for such a module.

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Representation Theory

Another class of simple graded Lie conformal algebras that cannot be embedded into general Lie conformal algebras

In a previous paper by the authors, we obtain the first example of a finitely freely generated simple Z -graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this paper, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying Z -graded simple Lie conformal algebras G= ????i=?? G i satisfying the following, (1) G 0 ?�Vir , the Virasoro conformal algebra; (2) Each G i for i?��?1 is a Vir -module of rank one. These algebras include some Lie conformal algebras of Block type.

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Representation Theory

Approximation of nilpotent orbits for simple Lie groups

We propose a systematic and topological study of limits lim ν??0 + G R ??νx) of continuous families of adjoint orbits for non-compact simple Lie groups. This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of SL n (R) and SU(p,q) are computed in detail.

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Representation Theory

Approximations of tilting modules

For any subset Θ of the natural numbers and any dominant weight λ we construct an indecomposable module S Θ (λ) with maximal weight λ for the quantum group U Z p , where Z p is a quantum version of the p -adic integers. For Θ=∅ we obtain the Weyl module, for Θ=N we obtain the indecomposable tilting module. Each S Θ (λ) admits a Weyl filtration, and for l∈Θ the character of S Θ (λ) equals the character of an (in general decomposable) tilting module for the quantum group at a primitive p l -th root of unity.

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Representation Theory

Arbitrarily large Morita Frobenius numbers

We construct blocks of finite groups with arbitrarily large Morita Frobenius numbers, an invariant which determines the size of the minimal field of definition of the associated basic algebra. This answers a question of Benson and Kessar. This also improves upon a result of the second author where arbitrarily large O -Morita Frobenius numbers are constructed.

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Representation Theory

Arthur packets for G 2 and perverse sheaves on cubics

This paper begins the project of defining Arthur packets of all unipotent representations for the p -adic exceptional group G 2 . Here we treat the most interesting case by defining and computing Arthur packets with component group S 3 . We also show that the distributions attached to these packets are stable, subject to a hypothesis. This is done using a self-contained microlocal analysis of simple equivariant perverse sheaves on the moduli space of homogeneous cubics in two variables. In forthcoming work we will treat the remaining unipotent representations and their endoscopic classification and strengthen our result on stability.

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Representation Theory

Axiomatizing Subcategories of Abelian Categories

We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find intrinsic axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting subcategories, and cluster tilting subcategories of abelian categories. As a consequence we prove that any d -abelian category is equivalent to a d -cluster tilting subcategory of an abelian category, without any assumption on the categories being projectively generated.

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Representation Theory

BPS Lie algebras and the less perverse filtration on the preprojective CoHA

We introduce a new perverse filtration on the Borel-Moore homology of the stack of representations of a preprojective algebra Π Q , by proving that the derived direct image of the dualizing mixed Hodge module along the morphism to the coarse moduli space is pure. We show that the zeroth piece of the resulting filtration on the preprojective CoHA is isomorphic to the universal enveloping algebra of the associated BPS Lie algebra g Π Q , and that the spherical Lie subalgebra of this algebra contains half of the Kac-Moody Lie algebra associated to the real subquiver of Q . Lifting g Π Q to a Lie algebra in the category of mixed Hodge modules on the coarse moduli space of Π Q -modules, we prove that the intersection cohomology of spaces of semistable Π Q -modules provide "cuspidal cohomology" for g Π Q - a conjecturally complete space of simple hyperbolic roots for this Lie algebra.

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