14 research outputs found
Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups
Let be an affine Weyl group, and let be a field of characteristic
. The diagrammatic Hecke category for over is a
categorification of the Hecke algebra for with rich connections to modular
representation theory. We explicitly construct a functor from to
a matrix category which categorifies a recursive representation , where is the rank of the
underlying finite root system. This functor gives a method for understanding
diagrammatic Soergel bimodules in terms of other diagrammatic Soergel bimodules
which are "smaller" by a factor of . It also explains the presence of
self-similarity in the -canonical basis, which has been observed in small
examples. By decategorifying we obtain a new lower bound on the -canonical
basis, which corresponds to new lower bounds on the characters of the
indecomposable tilting modules by the recent -canonical tilting character
formula due to Achar-Makisumi-Riche-Williamson.Comment: 62 pages, many figures, best viewed in colo
Path isomorphisms between quiver Hecke and diagrammatic Bott-Samelson endomorphism algebras
We construct an explicit isomorphism between (truncations of) quiver Hecke
algebras and Elias-Williamson's diagrammatic endomorphism algebras of
Bott-Samelson bimodules. As a corollary, we deduce that the decomposition
numbers of these algebras (including as examples the symmetric groups and
generalised blob algebras) are tautologically equal to the associated
-Kazhdan-Lusztig polynomials, provided that the characteristic is greater
than the Coxeter number. We hence give an elementary and more explicit proof of
the main theorem of Riche-Williamson's recent monograph and extend their
categorical equivalence to cyclotomic Hecke algebras, thus solving
Libedinsky-Plaza's categorical blob conjecture
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Semisimple filtrations of tilting modules for algebraic groups
Let be a reductive algebraic group over an algebraically closed field of characteristic . The indecomposable tilting modules for , which are labeled by highest weight, form an important class of self-dual representations over . In this thesis we investigate semisimple filtrations of minimal length (Loewy series) of tilting modules.
We first demonstrate a criterion for determining when tilting modules for arbitrary quasi-hereditary algebras are rigid, i.e. have a unique Loewy series. Our criterion involves checking that does not have certain subquotients whose composition factors extend more than one layer in the radical or socle series. We apply this criterion to show that the restricted tilting modules for are rigid when , something beyond the scope of previous work on this topic by Andersen and Kaneda.
Even when is not rigid, in many cases it has a particularly structured Loewy series which we call a balanced semisimple filtration, whose semisimple subquotients or "layers" are symmetric about some middle layer. Balanced semisimple filtrations also suggest a remarkably straightforward algorithm for calculating tilting characters from the irreducible characters. Applying Lusztig's character formula for the simple modules, we show that the algorithm agrees with Soergel's character formula for the regular indecomposable tilting modules for quantum groups at roots of unity. We then show that these filtrations really do exist for these tilting modules.
In the modular case, high weight tilting modules exhibit self-similarity in their characters at -power scales. This is due to what we call higher-order linkage, an old character-theoretic result relating modular tilting characters and quantum tilting characters at -power roots of unity. To better understand this behavior we describe an explicit categorification of higher-order linkage using the language of Soergel bimodules. Along the way we also develop the algebra and combinatorics of higher-order linkage at the de-categorified level. We hope that this will provide a foundation for a tilting character formula valid for all weights in the modular case
when is sufficiently large.This thesis was completed with the combined financial support of Trinity College (through an Internal Graduate Studentship) and the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge
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Radically filtered quasi-hereditary algebras and rigidity of tilting modules
AbstractLetAbe a quasi-hereditary algebra. We prove that in many cases, a tilting module is rigid (i.e. has identical radical and socle series) if it does not have certain subquotients whose composition factors extend more than one layer in the radical series or the socle series. We apply this theorem to show that the restricted tilting modules forSL4(K) are rigid, whereKis an algebraically closed field of characteristicp≥ 5.</jats:p
The modular Weyl-Kac character formula
We classify and explicitly construct the irreducible graded representations
of anti-spherical Hecke categories which are concentrated in one degree. Each
of these homogeneous representations is one-dimensional and can be
cohomologically constructed via a BGG resolution involving every (infinite
dimensional) standard representation of the category. We hence determine the
complete first row of the inverse parabolic -Kazhdan--Lusztig matrix for an
arbitrary Coxeter group and an arbitrary parabolic subgroup. This generalises
the Weyl--Kac character formula to all Coxeter systems (and their parabolics)
and proves that this generalised formula is rigid with respect to base change
to an arbitrary field
Quiver presentations and isomorphisms of Hecke categories and Khovanov arc algebras
We prove that the extended Khovanov arc algebras are isomorphic to the basic
algebras of anti-spherical Hecke categories for maximal parabolics of symmetric
groups. We present these algebras by quiver and relations and provide the full
submodule lattices of Verma modules
The anti-spherical Hecke categories for Hermitian symmetric pairs
We calculate the -Kazhdan--Lusztig polynomials for Hermitian symmetric
pairs and prove that the corresponding anti-spherical Hecke categories
categories are standard Koszul. We prove that the combinatorial invariance
conjecture can be lifted to the level of graded Morita equivalences between
subquotients of these Hecke categories