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Ruslan Maksimau

Publications and source records attributed to Ruslan Maksimau.

15 recordsLinked to original sources
arXiv2026

Qudit stabilizers beyond the free case and the twisted Kitaev model

We study the stabiliser formalism for qudits of arbitrary dimension $d$. In the free case, we show that the basic theorem of the stabiliser formalism remains valid: if the stabiliser subgroup $H$ is free as a $Z/dZ$-module and contains no non-trivial scalars, then the protected space $V^H$ is naturally identified with the state space of a smaller number of qudits of the same dimension, and the quotient $N(H)/H$ is identified with the Pauli group on a smaller number of qudits. We then remove the freeness assumption and describe the resulting structure in general. In this case, the protected space is identified with a tensor product of qudit spaces of possibly smaller dimensions, and the quotient $N(H)/H$ is described by a corresponding product of qudit Pauli groups, possibly of smaller dimensions, over a common center. We also characterise the shifted free case, which is exactly the situation in which $N(H)/H$ is again an ordinary qudit Pauli group. Our approach is algebraic and uniform, and applies in particular to the qudit Kitaev model and to its shifted and twisted variants.

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arXiv2025

Singularities in Calogero--Moser Varieties

In this article we describe completely the singularities appearing in Calogero--Moser varieties associated (at any parameter) to the wreath product symplectic reflection groups. We do so by parameterizing the symplectic leaves in the variety, describing combinatorially the resulting closure relation and computing a transverse slice to each leaf. We also show that the normalization of the closure of each symplectic leaf is isomorphic to a Calogero--Moser variety for an associated (explicit) subquotient of the symplectic reflection group. This confirms a conjecture of Bonnaf\'e for these groups. We use the fact that the Calogero--Moser varieties associated to wreath products can be identified with certain Nakajima quiver varieties. In particular, our result identifying the normalization of the closure of each symplectic leaf with another quiver variety holds for arbitrary quiver varieties.

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arXiv2025

Stratifying quiver Schur algebras via ersatz parity sheaves

We propose an extension of the theory of parity sheaves, which allows for non-locally constant sheaves along strata. Our definition is tailored for proving the existence of (proper, quasihereditary, etc) stratifications of $\mathrm{Ext}$-algebras. We use this to study quiver Schur algebras $A(\alpha)$ for the cyclic quiver of length $2$. We find a polynomial quasihereditary structure on $A(\alpha)$ compatible with the categorified PBW basis of McNamara and Kleshchev-Muth, and sharpen their results to arbitrary characteristic. We also prove that semicuspidal algebras of $A(n\delta)$ are polynomial quasihereditary covers of semicuspidal algebras of the corresponding KLR algebra $R(n\delta)$, and compute them diagrammatically.

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arXiv2024

Geometric categorifications of Verma modules: Grassmannian Quiver Hecke algebras

Naisse and Vaz defined an extension of KLR algebras to categorify Verma modules. We realise these algebras geometrically as convolution algebras in Borel-Moore homology. For this we introduce Grassmannian-Steinberg quiver flag varieties. They generalize Steinberg quiver flag varieties in a non-obvious way, reflecting the diagrammatics from the Naisse-Vaz construction. Using different kind of stratifications we provide geometric explanations of the rather mysterious algebraic and diagrammatic basis theorems. A geometric categorification of Verma modules was recently found in the special case of $\mathfrak{sl}_2$ by Rouquier. Rouquier's construction uses coherent sheaves on certain quasi-map spaces to flag varieties (zastavas), whereas our construction is implicitly based on perverse sheaves. Both should be seen as parts (on dual sides) of a general geometric framework for the Naisse-Vaz approach. We first treat the (substantially easier) $\mathfrak{sl}_2$ case in detail and construct as a byproduct a geometric dg-model of the nil-Hecke algebras. The extra difficulties we encounter in general require the use of more complicated Grassmannian-Steinberg quiver flag varieties. Their definition arises from combinatorially defined diagram varieties which we assign to each Naisse-Vaz basis diagram. Our explicit analysis here might shed some light on categories of coherent sheaves on more general zastava spaces studied by Feigin-Finkelberg-Kuznetsov-Mirković and Braverman, which we expect to occur in a generalization of Rouquier's construction away from $\mathfrak{sl}_2$.

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arXiv2023

DG-Enhanced Hecke and KLR Algebras

We construct DG-enhanced versions of the degenerate affine Hecke algebra and of the affine Hecke algebra. We extend Brundan-Kleshchev and Rouquier's isomorphism and prove that after completion DG-enhanced versions of affine Hecke algebras (degenerate or nondegenerate) are isomorphic to completed DG-enhanced versions of KLR algebras for suitably defined quivers. As a byproduct, we deduce that these DG-algebras have homologies concentrated in degree zero. These homologies are isomorphic respectively to the degenerate cyclotomic Hecke algebra and the cyclotomic Hecke algebra.

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arXiv2022

Symplectic leaves of Calogero-Moser spaces of type $G(\ell,1,n)$

We study symplectic leaves of Calogero-Moser spaces of type $G(\ell,1,n)$. We prove that the normalization of the closure of each symplectic leaf is isomorphic to some Calogero-Moser space. We also give a nice combinatorial parameterization of the symplectic leaves.

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arXiv2020

KLR and Schur algebras for curves and semi-cuspidal representations

Given a smooth curve $C$, we define and study analogues of KLR algebras and quiver Schur algebras, where quiver representations are replaced by torsion sheaves on $C$. In particular, they provide a geometric realization for certain affinized symmetric algebras. When $C=\mathbb P^1$, a version of curve Schur algebra turns out to be Morita equivalent to the imaginary semi-cuspidal category of the Kronecker quiver in any characteristic. As a consequence, we argue that one should not expect to have a reasonable theory of parity sheaves for affine quivers.

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arXiv2020

Categorical representations, KLR algebras and Koszul duality

The parabolic category $\mathcal O$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf A$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf A$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf A$ at level $-N-e$ can be decomposed in terms of components of the functor $F$ for the category $\mathbf A$ at level $-N-e-1$. To prove this, we use the approach of categorical representations. We prove a general fact about categorical representations: a category with an action of $\widetilde{\mathfrak sl}_{e+1}$ contains a subcategory with an action of $\widetilde{\mathfrak sl}_{e}$. To prove this claim, we construct an isomorphism between the KLR algebra associated with the quiver $A_{e-1}^{(1)}$ and a subquotient of the KLR algebra associated with the quiver $A_{e}^{(1)}$.

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arXiv2020

Affine category O, Koszul duality and Zuckerman functors

The parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf{A}$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf{A}$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf{A}$ at level $-N-e$ can be decomposed in terms of the components of the functor $F$ for the category $\mathbf{A}$ at level $-N-e-1$. To prove this, we use the following fact: a category with an action of $\widetilde{\mathfrak sl}_{e+1}$ contains a (canonically defined) subcategory with an action of $\widetilde{\mathfrak sl}_{e}$. We also prove a general statement that says that in some general situation a functor that satisfies a list of axioms is automatically Koszul dual to some sort of Zuckerman functor.

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arXiv2020

Fixed points in smooth Calogero-Moser spaces

We prove that every irreducible component of the fixed point variety under the action of $d$-th roots of unity in a smooth Caloger-Moser space is isomorphic to a Calogero-Moser space associated with another reflection group.

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arXiv2020

Higher level affine Schur and Hecke algebras

We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.

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arXiv2019

Flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology

We prove the conjecture that flag versions of quiver Grassmannians (also known as Lusztig's fibers) for Dynkin quivers (types $A$, $D$, $E$) have no odd cohomology groups over an arbitrary ring. Moreover, for types A and D we prove that these varieties have affine pavings. We also show that to prove the same statement for type E, it is enough to check this for indecomposable representations. We also give a flag version of the result of Cerulli Irelli-Esposito-Franzen-Reineke on rigid representations: we prove that flag versions of quiver Grassmannians for rigid representations have a diagonal decomposition. In particular, they have no odd cohomology groups.

math.RT↗
arXiv2019

Categorical representations and KLR algebras

We prove that the KLR algebra associated with the cyclic quiver of length $e$ is a subquotient of the KLR algebra associated with the cyclic quiver of length $e+1$. We also give a geometric interpretation of this fact. This result has an important application in the theory of categorical representations. We prove that a category with an action of $\widetilde{\mathfrak{sl}}_{e+1}$ contains a subcategory with an action of $\widetilde{\mathfrak{sl}}_{e}$. We also give generalizations of these results to more general quivers and Lie types.

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