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Register a new userCategorification via blocks of modular representations II
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Bernstein, Frenkel and Khovanov have constructed a categorification of tensor products of the standard representation of $mathfrak{sl}_2$ using singular blocks of category $mathcal{O}$ for $mathfrak{sl}_n$. In earlier work, we construct a positive characteristic analogue using blocks of representations of $mathfrak{sl}_n$ over a field $textbf{k}$ of characteristic $p > n$, with zero Frobenius character, and singular Harish-Chandra character. In the present paper, we extend these results and construct a categorical $mathfrak{sl}_k$-action, following Sussans approach, by considering more singular blocks of modular representations of $mathfrak{sl}_n$. We consider both zero and non-zero Frobenius central character. In the former setting, we construct a graded lift of these categorifications which are equivalent to a geometric construction of Cautis, Kamnitzer and Licata. We establish a Koszul duality between two geometric categorificatons constructed in their work, and resolve a conjecture of theirs. For non-zero Frobenius central characters, we show that the geometric approach to categorical symmetric Howe duality by Cautis and Kamnitzer can be used to construct a graded lift of our categorification using singular blocks of modular representations of $mathfrak{sl}_n$.
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Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories enriched with a p-differential which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider 2-categories stemming from bimodules over a p-dg category in detail. This class is of particular importance in the categorification of quantum groups, which allows us to apply our results to cyclotomic quotients of the categorifications of small quantum group of type $mathfrak{sl}_2$ at prime roots of unity by Elias-Qi [Advances in Mathematics 288 (2016)]. Passing to stable 2-representations gives a way to construct triangulated 2-representations, but our main focus is on working with p-dg enriched 2-representations that should be seen as a p-dg enhancement of these triangulated ones.
We introduce Brauer characters for representations of the bismash products of groups in characteristic p > 0, p not 2 and study their properties analogous to the classical case of finite groups. We then use our results to extend to bismash products a theorem of Thompson on lifting Frobenius-Schur indicators from characteristic p to characteristic 0
The core of a finite-dimensional modular representation $M$ of a finite group $G$ is its largest non-projective summand. We prove that the dimensions of the cores of $M^{otimes n}$ have algebraic Hilbert series when $M$ is Omega-algebraic, in the sense that the non-projective summands of $M^{otimes n}$ fall into finitely many orbits under the action of the syzygy operator $Omega$. Similarly, we prove that these dimension sequences are eventually linearly recursive when $M$ is what we term $Omega^{+}$-algebraic. This partially answers a conjecture by Benson and Symonds. Along the way, we also prove a number of auxiliary permanence results for linear recurrence under operations on multi-variable sequences.
In 2009, Keller and Yang categorified quiver mutation by interpreting it in terms of equivalences between derived categories. Their approach was based on Ginzburgs Calabi-Yau algebras and on Derksen-Weyman-Zelevinskys mutation of quivers with potential. Recently, Matthew Pressland has generalized mutation of quivers with potential to that of ice quivers with potential. In this paper, we show that his rule yields derived equivalences between the associated relative Ginzburg algebras, which are special cases of Yeungs deformed relative Calabi-Yau completions arising in the theory of relative Calabi-Yau structures due to Toen and Brav-Dyckerhoff. We illustrate our results on examples arising in the work of Baur-King-Marsh on dimer models and cluster categories of Grassmannians. We also give a categorification of mutation at frozen vertices as it appears in recent work of Fraser-Sherman-Bennett on positroid cluster structures.
We describe a categorification of the Double Affine Hecke Algebra ${mathcal{H}kern -.4emmathcal{H}}$ associated with an affine Lie algebra $widehat{mathfrak{g}}$, a categorification of the polynomial representation and a categorification of Macdonald polynomials. All categorification results are given in the derived setting. That is, we consider the derived category associated with graded modules over the Lie superalgera ${mathfrak I}[xi]$, where ${mathfrak I}subsetwidehat{mathfrak{g}}$ is the Iwahori subalgebra of the affine Lie algebra and $xi$ is a formal odd variable. The Euler characteristic of graded characters of a complex of ${mathfrak I}[xi]$-modules is considered as an element of a polynomial representation. First, we show that the compositions of induction and restriction functors associated with minimal parabolic subalgebras ${mathfrak{p}}_{i}$ categorify Demazure operators $T_i+1in{mathcal{H}kern -.4emmathcal{H}}$, meaning that all algebraic relations of $T_i$ have categorical meanings. Second, we describe a natural collection of complexes ${mathbb{EM}}_{lambda}$ of ${mathfrak I}[xi]$-modules whose Euler characteristic is equal to nonsymmetric Macdonald polynomials $E_lambda$ for dominant $lambda$ and a natural collection of complexes of $mathfrak{g}[z,xi]$-modules ${mathbb{PM}}_{lambda}$ whose Euler characteristic is equal to the symmetric Macdonald polynomial $P_{lambda}$. We illustrate our theory with the example $mathfrak{g}=mathfrak{sl}_2$ where we construct the cyclic representations of Lie superalgebra ${mathfrak I}[xi]$ such that their supercharacters coincide with renormalizations of nonsymmetric Macdonald polynomials.
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