?
Recent Developments in Representation Theory
Vol. 673.
Providence :
AMS, 2016.
Academic editor: A. Martsinkovsky, G. Todorov, K. Igusa
Compiler: A. Martsinkovsky
This volume contains the proceedings of the Maurice Auslander Distinguished Lectures and International Conference, held in April 2013 and April-May 2014, in Falmouth, MA.
Smirnov E., , in : Recent Developments in Representation Theory. Vol. 673.: Providence : AMS, 2016. P. 179-226.
The aim of these notes is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V. Kiritchenko, V. Timorin ...
Added: October 13, 2015
Keywords: representation theory
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
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
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
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
Tokyo : Mathematical Society of Japan, 2018
This volume is the proceedings of the conference "Representation Theory, Special Functions and Painlevé Equations" at the Research Institute for Mathematical Sciences, Kyoto University from March 3 to March 6 in 2015 ...
Added: October 8, 2019
Halacheva I., Kamnitzer J., Rybnikov L. G. et al., Duke Mathematical Journal 2020 Vol. 169 No. 12 P. 2337-2419
Fix a semisimple Lie algebra 𝔤. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for 𝔤-representations. These algebras depend on a parameter in the Deligne–Mumford moduli space of marked stable genus 0 curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the tensor product multiplicity space and gives us a ...
Added: July 22, 2020
Smilga I., / Cornell University. Series arXiv "math". 2018. No. 1802.07193.
We prove a partial converse to the main theorem of the author's previous paper "Proper affine actions: a sufficient criterion" (submitted; available at arXiv:1612.08942). More precisely, let G be a semisimple real Lie group with a representation rho on a finite-dimensional real vector space V, that does not satisfy the criterion from the previous paper. ...
Added: September 26, 2018
Leonid Rybnikov, International Mathematics Research Notices 2018 No. 1 P. 202-235
Cactus group is the fundamental group of the real locus of the Deligne–Mumford moduli space of stable rational curves. This group appears naturally as an analog of the braid group in coboundary monoidal categories. We define an action of the cactus group on the set of Bethe vectors of the Gaudin magnet chain corresponding to ...
Added: February 6, 2018
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
Bufetov A., Gorin V., / Cornell University. Series math "arxiv.org". 2013. No. 1311.5780.
We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the ...
Added: December 4, 2013
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
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
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
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
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
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
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
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
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
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
Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189
The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...
Added: January 28, 2020