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## Characters of Feigin-Stoyanovsky subspaces and Brion's theorem

Functional Analysis and Its Applications. 2015. Vol. 49. No. 1. P. 15-24.

Makhlin I.

We give an alternative proof of the main result of [1]; the proof relies on Brion’s theorem about convex polyhedra. The result itself can be viewed as a formula for the character of the Feigin-Stoyanovsky subspace of an integrable irreducible representation of the affine Lie algebra widehatsln(C). Our approach is to assign integer points of a certain polytope to vectors comprising a monomial basis of the subspace and then compute the character by using (a variation of) Brion’s theorem.

Language:
English

Feigin E., Makhlin I., / Cornell University. Series math "arxiv.org". 2016. No. arXiv:1604.08844.

FFLV polytopes describe monomial bases in irreducible representations of sln. We study various sets of vertices of FFLV polytopes. First, we prove the locality of set of vertices with respect to the type A Dynkin diagram. Second, we describe all the permutation vertices. Third, we describe all the simple vertices and prove that their number ...

Added: May 6, 2016

Feigin B. L., Функциональный анализ и его приложения 2014 № 3

We study commutative vertex operator algebras. These algebras are isomorphic to certain subalgebras in Kac-Moody vertex operator algebras. We describe systems of relations and degenerations to quadratic algebras. Our approach leads to the fermionic formulas for characters. ...

Added: April 14, 2014

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

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

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

Providence : AMS, 2016

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. ...

Added: October 13, 2015

Ayzenberg A., Масуда М., Сато Т., Труды Математического института им. В.А. Стеклова РАН 2022 Т. 317 С. 5-26

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 ...

Added: October 27, 2022

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

Khoroshkin S. M., Nazarov M., , in : Advanced Studies in Pure Mathematics (т.76 Representations Theory, Special Functions and Painleve Equations - RIMS 2015). Vol. 76.: Tokyo : Mathematical Society of Japan, 2018. Ch. 8. P. 275-302.

Arakawa, Suzuki and Tsuchiya defined a correspondence between certain modules of the trigonometric Cherednik algebra CN depending on a parameter κ∈C, and certain modules of the affine Lie algebra slˆm of level κ−m. We give a detailed proof of this correspondence by working with the affine Lie algebra glˆm alongside of slˆm. We also relate ...

Added: October 8, 2019

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

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

Feigin E., Makedonskyi I., / Cornell University. Series arXiv "math". 2015. No. 1507.01362.

The main goal of our paper is to establish a connection between the Weyl modules of the current Lie superalgebras (twisted and untwisted) attached to osp(1,2) and the nonsymmetric Macdonald polynomials of types $A_2^2$ and ${A_2}^{2\dagger}$ . We compute the dimensions and construct bases of the Weyl modules. We also derive explicit formulas for the ...

Added: July 8, 2015

Feigin B. L., Makhlin I., / Cornell University. Series math "arxiv.org". 2015.

We present a new combinatorial formula for Hall-Littlewood functions associated with the affine root system of type A~n−1, i.e. corresponding to the affine Lie algebra slˆn. Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation.Our formula can ...

Added: August 7, 2015

Dumanski I., Feigin E., Finkelberg M., Forum of Mathematics, Sigma 2021 Vol. 9

We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson-Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules over the current Lie algebra. ...

Added: September 8, 2021

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

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

Braverman A., Rybnikov L. G., Feigin B. L. et al., Communications in Mathematical Physics 2011 Vol. 308 No. 2 P. 457-478

Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P^2. More precisely, it predicts ...

Added: May 12, 2012

Vladimir L. Popov, , in : Lie Groups, Geometry, and Representation Theory. A Tribute to the Life and Work of Bertram Kostant. Vol. 326.: Copyright Holder Springer Nature Switzerland AG, 2018. Ch. 16. P. 459-479.

We first establish several general properties of modality of algebraic group
actions. In particular,we introduce the notion of a modality-regular action and prove
that every visible action is modality-regular. Then, using these results, we classify irreducible
linear representations of connected simple algebraic groups of every fixed
modality < 3. Next, exploring a finer geometric structure of linear actions, we ...

Added: October 27, 2018

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

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