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## Вторые когомологии регулярных полупростых многообразий Хессенберга на основе ГКМ-теории

Труды Математического института им. В.А. Стеклова РАН. 2022. Т. 317. С. 5-26.

Ayzenberg A., Масуда М., Сато Т.

We describe the second cohomology of a regular semisimple Hessenberg variety by generators and relations explicitly in terms of GKM theory. The cohomology of a regular semisimple Hessenberg variety becomes a module of a symmetric group Sn by the dot action introduced by Tymoczko. As an application of our explicit description, we give a formula describing the isomorphism class of the second cohomology as an Sn-module. Our formula is not exactly the same as the known formula by Chow or Cho, Hong, and Lee, but they are equivalent. We also discuss its higher degree generalization.

Feigin E., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2014. No. 1407.6316.

The Cherednik-Orr conjecture expresses the t\to\infty limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove this conjecture in several special cases. ...

Added: August 10, 2014

Positselski L., Arkhipov S., Rumynin D., Basel : Birkhauser/Springer, 2010

We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories ...

Added: March 19, 2013

Feigin E., Makedonskyi I., Orr D., Advances in Mathematics 2018 Vol. 330 P. 997-1033

We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric q-Whittaker functions. In particular, we show that the series part of the nonsymmetric q-Whittaker function is a generating function for the graded characters of generalized global Weyl modules. ...

Added: September 13, 2018

Makhlin I., Selecta Mathematica, New Series 2015

We exploit the idea that the character of an irreducible finite dimensional $\mathfrak{gl}_n$-module is the sum of certain exponents of integer points in a Gelfand-Tsetlin polytope and can thus be calculated via Brion's theorem. In order to show how the result of such a calculation matches Weyl's character formula we prove some interesting combinatorial traits ...

Added: September 29, 2014

Antipov M., Звонарёва А. О., Journal of Mathematical Sciences 2014 Vol. 202 No. 3 P. 333-345

In this paper, all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 are described, using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application, all two-term tilting complexes over a Brauer star algebra are described and their endomorphism rings are ...

Added: December 25, 2018

Braverman A., Dobrovolska G., Michael Finkelberg, / Cornell University. Series math "arxiv.org". 2014.

Let G be an almost simple simply connected group over complex numbers. For a positive element α of the coroot lattice of G let Z^α denote the space of based maps from the projective line to the flag variety of G of degree α. This space is known to be isomorphic to the space of ...

Added: February 3, 2015

Braverman A., Michael Finkelberg, / Cornell University. Series math "arxiv.org". 2014.

In this note, we extend the results of arxiv:1111.2266 and arxiv:1203.1583 to the non simply laced case. To this end we introduce and study the twisted zastava spaces. ...

Added: February 5, 2015

Rovinsky M., / Cornell University. Series math "arxiv.org". 2014.

In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite field) are studied. Many results here are well-known to the experts, at least in the case of {\sl linear representations} of ...

Added: September 17, 2014

Positselski L., / Cornell University. Series math "arxiv.org". 2014. No. 1404.5011.

We construct the reduction of an exact category with a twist functor with respect to an element of its graded center in presence of an exact-conservative forgetful functor annihilating this central element. The procedure allows, e.g., to recover the abelian/exact category of modular representations of a finite group from the exact category of its l-adic ...

Added: April 22, 2014

Feigin B. L., Miwa T., Jimbo M. et al., / Cornell University Library. 2013. No. 1309.2147.

We construct an analog of the subalgebra $U\mathfrak{gl}(n)\otimes U\mathfrak{gl}(m)\subset U\mathfrak{gl}(m+n)$ in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra. ...

Added: April 24, 2014

Alexander I. Efimov, / Cornell University. Series math "arxiv.org". 2014.

In this paper we study the derived categories of coherent sheaves on Grassmannians Gr(k,n), defined over the ring of integers. We prove that the category D^b(Gr(k,n)) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of GL_k. This in particular implies existence of a full exceptional collection, which is ...

Added: February 2, 2015

Michael Finkelberg, Schechtman V., / Cornell University. Series math "arxiv.org". 2014.

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

Added: January 30, 2015

Cruz Morales J. A., Galkin S., / Cornell University. Series math "arxiv.org". 2013. No. 1301.4541.

In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. ...

Added: May 27, 2013

Braverman A., Michael Finkelberg, Nakajima H., / Cornell University. Series math "arxiv.org". 2014.

We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincar\'e pairing) in terms of representation theory of some vertex operator algebras ("W-algebras"). ...

Added: January 30, 2015

Levin A., Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2014.

We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard
(KZB) equations with $\tilde N$ punctures by deformation of the corresponding
quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit
of the first one brings the model to the ordinary rational KZ equation. Another
one is $\tau$. At the level of classical mechanics the deformation parameter
$\tau$ allows to extend the ...

Added: January 23, 2015

Brav C. I., Thomas H., Mathematische Annalen 2011 Vol. 351 No. 4 P. 1005-1017

We establish faithfulness of braid group actions generated by twists along an ADE configuration of 22-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity. ...

Added: September 29, 2014

Bezrukavnikov R., Finkelberg M. V., / Cornell University. Series math "arxiv.org". 2012. No. 1208.3696.

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert ...

Added: February 6, 2013

Bershtein M., Gavrylenko P., Marshakov A., / arXiv.org. Series arXiv.org "hep-th". 2017. No. 1705.00957.

We study twist-field representations of the W-algebras and generalize the construction of the corresponding vertex operators to D- and B-series. We demonstrate how the computation of characters of such representations leads to the nontrivial identities involving lattice theta-functions. We propose a construction of their exact conformal blocks, which for D-series express them in terms of ...

Added: May 4, 2017

Cherepanov V., Математический сборник 2019

We consider effective actions of a compact torus Tn−1 on an even-dimensional smooth manifold M2n with isolated fixed points. We prove that under certain conditions on weights of tangent representations, the orbit space is a manifold with corners. Given that the action is Hamiltonian, the orbit space is homeomorphic to Sn+1∖(U1⊔…⊔Ul) where Sn+1 is the (n+1)--sphere and U1,…,Ul are open domains. We apply the results to ...

Added: October 28, 2020

Gorsky Evgeny, Selecta Mathematica, New Series 2013 Vol. 19 No. 1 P. 125-140

A theorem of Y. Berest, P. Etingof and V. Ginzburg states that finite-dimensional irreducible representations of a type A rational Cherednik algebra are classified by one rational number m/n. Every such representation is a representation of the symmetric group Sn . We compare certain multiplicity spaces in its decomposition into irreducible representations of Sn with the spaces of differential forms ...

Added: December 9, 2014

Switzerland : Birkhauser/Springer, 2022

The conference “Interactions between Representation Theory and Algebraic Geometry” was held at the University of Chicago on August 21–25, 2017. It brought together about 150 participants from several major universities in the USA and abroad, more than half of whom were junior mathematicians. It featured 21 talks by eminent mathematicians from the USA, Europe, and Asia on topics ...

Added: June 20, 2022

Shirokov D., Marchuk N., Advances in Applied Clifford Algebras 2008 Vol. 18 No. 2 P. 237-254

For the complex Clifford algebra <img /> (p, q) of dimension n = p + q we define a Hermitian scalar product. This scalar product depends on the signature (p, q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These ...

Added: June 16, 2015

Feigin E., Littelmann P., Fourier G., , in : Symmetries, Integrable Systems and Representations. Vol. 40: Symmetries, Integrable Systems and Representations.: Springer, 2013. P. 35-63.

We study the PBW-filtration on the highest weight representations V(λ) of the Lie algebras of type A n and C n . This filtration is induced by the standard degree filtration on . In previous papers, the authors studied the filtration and the associated graded algebras and modules over the complex numbers. The aim of ...

Added: February 5, 2013

Burman Y. M., / Cornell University. Series math "arxiv.org". 2013. No. 1309.4477.

Given a representation V of a group G, there are two natural ways of defining a representation of the group algebra k[G] in the external power V^{\wedge m}. The set L(V) of elements of k[G] for which these two ways give the same result is a Lie algebra and a representation of G. For the ...

Added: November 19, 2013