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Gleb Koshevoy

Publications and source records attributed to Gleb Koshevoy.

At least 19 recordsLinked to original sources
arXiv2026

Hodge numbers for orbifolds of Calabi-Yau threefolds Fermat type and the Roan pairs

First, for orbifolds of Calabi-Yau threefolds of Fermat type, we define Roan's Hodge numbers. We prove, Theorem \ref{main} , that for all orbifols of Calabi-Yau threefolds Fermat type, Roan's Hodge numbers correctly count the stringy Euler numbers due to Vafa formula. Second, for Calabi-Yau threefolds Fermat type, we apply Roan's Hodge numbers to get a relation between the Borcea-Voisin construction \cite{Borcea, Voisin} and Berglund-H\"ubsch-Krawits mirror symmetry (BHK mirror symmetry). Third, for orbifolds of K3 surfaces Fermat type, we establish relations twisted and untwisted parts of the second mixed cohomology to deformations and the Roan pairs. This allow us to confirm the BHK mirror symmetry for such orbifolds, since the Euler numbers computed by the Vafa formula are equal $24$ for all of them.

hep-th↗
arXiv2026

Quaternities, correspondences, and tetrahedron equations (Summa tetralogiae)

The aim of this note is: (a) to propose a generalization of tetrahedron equations from \cite{S} and of their solutions. Due to appearance of a larger number of parameters the $R$-matrices from \cite{S} will be replaced by "$R$-correspondences". (b) To rephrase these equations in terms of Wronskian evolutions in the spirit of \cite{SV}. (c) To discuss some elementary structures of cohomological flavour lying behind our considerations. We call them "quaternities", or "bibitorsors"; they might be not without an independent interest.

math.RA↗
arXiv2025

Bruhat operads

We describe some planar operads built from the higher Bruhat orders and show that they admit a multiplication.

math.CO↗
arXiv2025

Products of Kirillov-Reshetikhin modules and maximal green sequences

We show that a $q$-character of a Kirillov-Reshetikhin module (KR modules) for untwisted quantum affine algebras of simply laced types $A_n^{(1)}$, $D_n^{(1)}$, $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$ might be obtained from a specific cluster variable of a seed obtained by applying a maximal green sequence to the initial (infinite) quiver of the Hernandez-Leclerc cluster algebra. For a collection of KR-modules with nested supports, we show an explicit construction of a cluster seed, which has cluster variables corresponding to the $q$-characters of KR-modules of such a collection. We prove that the product of KR-modules of such a collection is a simple module. We also construct cluster seeds with cluster variables corresponding to $q$-characters of KR-modules of some non-nested collections. We make a conjecture that tensor products of KR-modules for such non-nested collections are simple. We show that the cluster Donaldson-Thomas transformations for double Bruhat cells for $ADE$ types can be computed using $q$-characters of KR-modules.

math.RT↗
arXiv2022

An algorithm for Berenstein-Kazhdan decoration functions and trails for classical Lie algebras

For a simply connected connected simple algebraic group $G$, it is known that a variety $B_{w_0}^-:=B^-\cap U\overline{w_0}U$ has a geometric crystal structure with a positive structure $θ^-_{\mathbf{i}}:(\mathbb{C}^{\times})^{l(w_0)}\rightarrow B_{w_0}^-$ for each reduced word $\mathbf{i}$ of the longest element $w_0$ of Weyl group. A rational function $Φ^h_{BK}=\sum_{i\in I}Δ_{w_0Λ_i,s_iΛ_i}$ on $B_{w_0}^-$ is called a half-potential, where $Δ_{w_0Λ_i,s_iΛ_i}$ is a generalized minor. Computing $Φ^h_{BK}\circ θ^-_{\mathbf{i}}$ explicitly, we get an explicit form of string cone or polyhedral realization of $B(\infty)$ for the finite dimensional simple Lie algebra $\mathfrak{g}={\rm Lie}(G)$. In this paper, for an arbitrary reduced word $\mathbf{i}$, we give an algorithm to compute the summand $Δ_{w_0Λ_i,s_iΛ_i}\circ θ^-_{\mathbf{i}}$ of $Φ^h_{BK}\circ θ^-_{\mathbf{i}}$ in the case $i\in I$ satisfies that for any weight $μ$ of $V(-w_0Λ_i)$ and $t\in I$, it holds $\langle h_t,μ\rangle\in\{2,1,0,-1,-2\}$. In particular, if $\mathfrak{g}$ is of type ${\rm A}_n$, ${\rm B}_n$, ${\rm C}_n$ or ${\rm D}_n$ then all $i\in I$ satisfy this condition so that one can completely calculate $Φ^h_{BK}\circ θ^-_{\mathbf{i}}$. We will also prove that our algorithm works in the case $\mathfrak{g}$ is of type ${\rm G}_2$.

math.QA↗
arXiv2022

The purity phenomenon for symmetric separated set-systems

Let $n$ be a positive integer. A collection $\cal S$ of subsets of $[n]=\{1,\ldots,n\}$ is called {\it symmetric} if $X\in {\cal S}$ implies $X^\ast\in {\cal S}$, where $X^\ast:=\{i\in [n]\colon n-i+1\notin X\}$. We show that in each of the three types of separation relations: {\it strong}, {\it weak} and {\it chord} ones, the following "purity phenomenon" takes place: all inclusion-wise maximal symmetric separated collections in $2^{[n]}$ have the same cardinality. These give "symmetric versions" of well-known results on the purity of usual strongly, weakly and chord separated collections of subsets of $[n]$, and in the case of weak separation, this extends a recent result due to Karpman on the purity of symmetric weakly separated collections in $\binom{[n]}{n/2}$ for $n$ even.

math.CO↗
arXiv2021

An algorithm for Berenstein-Kazhdan decoration functions and trails for minuscule representations

For a simply connected connected simple algebraic group $G$, a cell $B_{w_0}^-=B^-\cap U\overline{w_0}U$ is a geometric crystal with a positive structure $θ_{\textbf{i}}^-:(\mathbb{C}^{\times})^{l(w_0)}\rightarrow B_{w_0}^-$. Applying the tropicalization functor to a rational function $Φ^h_{BK}=\sum_{i\in I}Δ_{w_0Λ_i,s_iΛ_i}$ called the half decoration on $B_{w_0}^-$, one can realize the crystal $B(\infty)$ in $\mathbb{Z}^{l(w_0)}$. By computing $Φ^h_{BK}$, we get an explicit form of $B(\infty)$ in $\mathbb{Z}^{l(w_0)}$. In this paper, we give an algorithm to compute $Δ_{w_0Λ_i,s_iΛ_i}\circ θ_{\textbf{i}}^-$ explicitly for $i\in I$ such that $V(Λ_i)$ is a minuscule representation of $\mathfrak{g}={\rm Lie}(G)$. In particular, the algorithm works for all $i\in I$ if $\mathfrak{g}$ is of type ${\rm A}_n$. The algorithm computes a directed graph $DG$, called a decoration graph, whose vertices are labelled by all monomials in $Δ_{w_0Λ_i,s_iΛ_i}\circ θ_{\textbf{i}}^-(t_1,\cdots,t_{l(w_0)})$. The decoration graph has some properties similar to crystal graphs of minuscule representations. We also verify that the algorithm works in some other cases, for example, the case $\mathfrak{g}$ is of type ${\rm G}_2$ though $V(Λ_i)$ is non-minuscule.

math.QA↗
arXiv2021

Periods of the multiple Berglund-Huebsch-Krawitz mirrors

We consider the multiple Calaby-Yau (CY) mirror phenomenon which appears in Berglund-Hübsch-Krawitz (BHK) mirror symmetry. We show that for any pair of Calabi--Yau orbifolds that are BHK mirrors of a loop--chain type pair of Calabi--Yau manifolds in the same weighted projective space the periods of the holomorphic nonvanishing form coincide.

hep-th↗
arXiv2020

Majority rule on rhombus tilings and Condorcet super-domains

In this paper we consider a Condorcet domain (CD) formed by a rhombus tiling as a voting design and consider a problem of aggregation of voting designs using majority rule. A Condorcet super-domain is a collection of CDs obtained from rhombus tilings on a zonogone Z(n; 2) with the property that if voting designs (ballots) belong to this collection, then the simple majority rule does not yield cycles. A study of Condorcet super-domains and methods of constructing them form the main subject of this paper.

math.CO↗
arXiv2020

Lusztig polytopes and FFLV polytopes

In this paper we prove that in type $\tt A_n$, the Feigin-Fourier-Littelmann-Vinberg (FFLV) polytope coincides with the Minkowski sum of Lusztig polytopes arising from various reduced decompositions. Using this result, we formulate a conjecture about the crystal structures on FFLV polytopes.

math.RT↗
arXiv2019

Combinatorics of canonical bases revisited: String data in type $A$

We give a formula for the crystal structure on the integer points of the string polytopes and the $*$-crystal structure on the integer points of the string cones of type $A$ for arbitrary reduced words. As a byproduct we obtain defining inequalities for Nakashima-Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara $*$-involution on string data for a special choice of reduced word.

math.RT↗
arXiv2017

Monotone bargaining is Nash-solvable

Given two finite ordered sets $A = \{a_1, \ldots, a_m\}$ and $B = \{b_1, \ldots, b_n\}$, introduce the set of $m n$ outcomes of the game $O = \{(a, b) \mid a \in A, b \in B\} = \{(a_i, b_j) \mid i \in I = \{1, \ldots, m\}, j \in J = \{1, \ldots, n\}$. Two players, Alice and Bob, have the sets of strategies $X$ and $Y$ that consist of all monotone non-decreasing mappings $x: A \rightarrow B$ and $y: B \rightarrow A$, respectively. It is easily seen that each pair $(x,y) \in X \times Y$ produces at least one {\em deal}, that is, an outcome $(a,b) \in O$ such that $x(a) = b$ and $y(b) = a$. Denote by $G(x,y) \subseteq O$ the set of all such deals related to $(x,y)$. The obtained mapping $G = G_{m,n}: X \times Y \rightarrow 2^O$ is a game correspondence. Choose an arbitrary deal $g(x,y) \in G(x,y)$ to obtained a mapping $g : X \times Y \rightarrow O$, which is a game form. We will show that each such game form is tight and, hence, Nash-solvable, that is, for any pair $u = (u_A, u_B)$ of utility functions $u_A : O \rightarrow \mathbb R$ of Alice and $u_B: O \rightarrow \mathbb R$ of Bob, the obtained monotone bargaining game $(g, u)$ has at least one Nash equilibrium in pure strategies. Moreover, the same equilibrium can be chosen for all selections $g(x,y) \in G(x,y)$. We also obtain an efficient algorithm that determines such an equilibrium in time linear in $m n$, although the numbers of strategies $|X| = \binom{m+n-1}{m}$ and $|Y| = \binom{m+n-1}{n}$ are exponential in $m n$. Our results show that, somewhat surprising, the players have no need to hide or randomize their bargaining strategies, even in the zero-sum case.

math.CO↗
arXiv2017

Polyhedral parametrizations of canonical bases & cluster duality

We establish the relation of the potential function constructed by Gross-Hacking-Keel-Kontsevich's and Berenstein-Kazhdan's decoration function on the open double Bruhat cell in the base affine space $G/\mathcal{N}$ of a simple, simply connected, simply laced algebraic group $G$. As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on $G/\mathcal{N}$ arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations.

math.RT↗
arXiv2017

Combinatorics of canonical bases revisited: Type A

We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula" to compute the actions of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein-Fomin-Zelevinsky we prove an enhanced version of the Anderson-Mirković conjecture for the crystal structure on MV polytopes. We establish a duality between Kashiwara's string and Lusztig's parametrization, revealing that each of them is controlled by the crystal structure of the other. We identify the potential functions of the unipotent radical of $SL_n$ defined by Berenstein-Kazhdan and Gross-Hacking-Keel-Kontsevich, respectively, with a function arising from the crystal structure on Lusztig data.

math.RT↗