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Vadim Schechtman

Publications and source records attributed to Vadim Schechtman.

At least 19 recordsLinked to original sources
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

Pentagramma Mirificum. II

This article is a continuation of my previous paper on miraculous pentagrams published in Annales Scientifiques de Facult\'e des Sciences de Toulouse in 2013. We discuss the moduli space of miraculous pentagrams which is the Del Pezzo surface of degree 5, and present some examples related to Fibonacci numbers. We discuss also some relations between Fermat ascent and Poncelet theorem.

math.AG↗
arXiv2025

Bruhat operads

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

math.CO↗
arXiv2024

The Langlands formula and perverse sheaves

For a complex reductive Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$ and Weyl group $W$ we consider the category $\text{Perv}(W \backslash \mathfrak{h})$ of perverse sheaves on $W \backslash \mathfrak{h}$ smooth w.r.t. the natural stratification. We construct a category $\boldsymbol{\mathcal{C}}$ such that $\text{Perv}(W\backslash \mathfrak{h})$ is identified with the category of functors from $\boldsymbol{\mathcal{C}}$ to vector spaces. Objects of $\boldsymbol{\mathcal{C}}$ are labelled by standard parabolic subalgebras in $\mathfrak{g}$. It has morphisms analogous to the operations of parabolic induction (Eisenstein series) and restriction (constant term) of automorphic forms. In particular, the Langlands formula for the constant term of an Eisenstein series has a counterpart in the form of an identity in $\boldsymbol{\mathcal{C}}$. We define $\boldsymbol{\mathcal{C}}$ as the category of $W$-invariants (in an appropriate sense) in the category $Q$ describing perverse sheaves on $\mathfrak{h}$ smooth w.r.t. the root arrangement. This matches, in an interesting way, the definition of $W \backslash \mathfrak{h}$ itself as the spectrum of the algebra of $W$-invariants.

math.RT↗
arXiv2024

Homotopy chiral algebras

We define a notion of a homotopy chiral algebra (HCA), which means a chiral algebra up to higher homotopies, and prove that the Cech complex of a sheaf of chiral algebras admits a structure of a HCA.

math.AT↗
arXiv2023

N-spherical functors and categorification of Euler's continuants

Euler's continuants are universal polynomials expressing the numerator and denominator of a finite continued fraction whose entries are independent variables. We introduce their categorical lifts which are natural complexes (more precisely, coherently commutative cubes) of functors involving compositions of a given functor and its adjoints of various orders, with the differentials built out of units and counits of the adjunctions. In the stable infinity-categorical context these complexes/cubes can be assigned totalizations which are new functors serving as higher analogs of the spherical twist and cotwist. We define N-spherical functors by vanishing of the twist and cotwist of order N-1 in which case those of order N-2 are equivalences. The usual concept of a spherical functor corresponds to N=4. We characterize N-periodic semi-orthogonal decompositions of triangulated (stable infinity-) categories in terms of N-sphericity of their gluing functors. The procedure of forming iterated orthogonals turns out to be analogous to the procedure of forming a continued fraction.

math.CT↗
arXiv2022

De Rham - Witt KZ equations

We propose a de Rham - Witt version of the derived Knizhnik-Zamolodchikov equations, and of their hypergeometric realizations. We also propose de Rham - Witt versions of some classical theorems related to arbitrary hyperplane arrangements.

math-ph↗
arXiv2022

Chiral Janus complexes

We propose chiral analogues of some infinite complexes appearing in the description of the coherent derived categories for projective spaces.

math.AG↗
arXiv2022

PROBs and perverse sheaves II. Ran spaces and 0-cycles with coefficients

We consider the space Z(C,L) of 0-cycles on the complex line C with coefficients in a commutative monoid L subject to certain conditions. Such spaces include the symmetric products (for L=Z_+) and the Ran space (for L=T={ True, False} being the Boolean algebra of truth values). We describe the appropriately defined category of perverse sheaves on Z(C,L) in terms of the braided category (PROB) generated by the components of the universal $L$-graded bialgebra. We give another description in terms of so-called Janus sheaves which are objects of mixed functoriality (data covariant in one direction and contravariant in the other) on a category formed by certain matrices with entries in L. The matrices in question are analogs of contingency tables familiar in statistics.

math.CT↗
arXiv2022

Microlocal approach to Lusztig's symmetries

We reformulate the De Concini -- Toledano Laredo conjecture about the monodromy of the Casimir connection in terms of a relation between Lusztig's symmetries of quantum group modules and the monodromy in the vanishing cycles of factorizable sheaves.

math.AG↗
arXiv2022

Fourier transform on hyperplane arrangements

We consider the category of perverse sheaves on a complex vector space smooth with respect to a stratification given by an arrangement of hyperplanes with real equations. As shown in an earlier wotk of two of the authors, this category can be described in terms of certain diagrams of vector spaces labelled by all the faces of the real arrangement (we call such diagrams hyperbolic sheaves). In this paper we calculate, in these terms, several fundamental operations of sheaf theory such as forming the space of vanishing cycles, specialization and the Fourier-Sato transform.

math.AT↗
arXiv2022

Kostka numbers and Fourier duality

We relate the Fourier transform of perverse sheaves smooth along the coordinate hyperplane configuration in a complex vector space to the Deligne-Lusztig duality of unipotent representations of a general linear group over a finite field. A similar relation is established for arbitrary finite Coxeter groups.

math.RT↗
arXiv2021

Parabolic induction and perverse sheaves on h/W

For a complex reductive Lie group G with Lie algebra g, Cartan subalgebra h and Weyl group W, we describe the category of perverse sheaves on h/W smooth w.r.t the natural stratification. The answer is given in terms of mixed Bruhat sheaves, which are certain mixed sheaf-cosheaf data on cells of a natural cell decomposition of h/W. Using the parabolic Bruhat decomposition, we relate mixed Bruhat sheaves with the properties of various procedures of parabolic induction and restriction that connect different Levi subgroups in G.

math.AT↗
arXiv2021

Spherical adjunctions of stable $\infty$-categories and the relative S-construction

We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable $\infty$-categories. Building on this, we study the relative Waldhausen S-construction $S_\bullet(F)$ of a spherical functor $F$ and equip it with a natural paracyclic structure (``rotational symmetry''). This fulfills a part of the general program to provide a rigorous account of perverse schobers which are (thus far conjectural) categorifications of perverse sheaves. Namely, in terms of our previous identification of perverse sheaves on Riemann surfaces with Milnor sheaves, the relative $S$-construction with its paracyclic symmetry amounts to a categorification of the stalks of a Milnor sheaf at a singularity of the corresponding perverse sheaf. The action of the paracyclic rotation is a categorical analog of the monodromy on the vanishing cycles of a perverse sheaf. Having this local categorification in mind, we may view the S-construction of a spherical functor as defining a schober locally at a singularity. Each component $S_n(F)$ can be interpreted as a partially wrapped Fukaya category of the disk with coefficients in the schober and with $n+1$ stops at the boundary.

math.AG↗
arXiv2021

Lorentz groups of cyclotomic extensions

In this (mostly historical) note we show how a unified Kummer-Artin-Schreier sequence from [W], [SOS] may be recovered from the relativistic velocity addition law.

math.NT↗