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Register a new userCategorification of ice quiver mutation
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Yilin Wu
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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.
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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.
This is a concise introduction to Fomin-Zelevinskys cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition of cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster variables and rigid indecomposable objects of the cluster category.
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 $mathfrak{sl}_n$.
We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring $A_q(mathfrak{n}(w))$, associated with a symmetric Kac-Moody algebra and its Weyl group element $w$, admits a monoidal categorification via the representations of symmetric Khovanov-Lauda- Rouquier algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification, where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed of $A_q(mathfrak{n}(w))$ which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power of $q^{1/2}$. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.
We introduce and investigate new invariants on the pair of modules $M$ and $N$ over quantum affine algebras $U_q(mathfrak{g})$ by analyzing their associated R-matrices. From new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable $U_q(mathfrak{g})$-modules to become a monoidal categorification of a cluster algebra.
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