?
Generalized Weyl modules, alcove paths and Macdonald polynomials
Selecta Mathematica, New Series. 2017. Vol. 23. No. 4. P. 2863-2897.
Feigin E., Makedonskyi I.
Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum Bruhat graph. We make use of the Orr–Shimozono formula in order to prove that the $t=\infty$ specializations of the nonsymmetric Macdonald polynomials are equal to the characters of certain generalized Weyl modules.
Feigin E., Makedonskyi I., Orr D., / Cornell University. Series math "arxiv.org". 2016. No. 1605.01560.
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: May 6, 2016
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 E., Cherednik I., / Cornell University. Series math "arxiv.org". 2013. No. arXiv:1306.3146.
Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials, also called E-polynomails, in the limit t=infinity and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to ...
Added: June 24, 2013
Feigin E., Kato S., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2017. No. 1703.04108.
We study the nonsymmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the nonsymmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on ...
Added: March 20, 2017
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
Feigin E., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2017. No. 1709.05674.
The goal of this paper is twofold. First, we write down the semi-infinite Pl\"ucker relations, describing the Drinfeld-Pl\"ucker embedding of the (formal version of) semi-infinite flag varieties in type A. Second, we study the homogeneous coordinate ring, i.e. the quotient by the ideal generated by the semi-infinite Pl\"ucker relations. We establish the isomorphism with the ...
Added: September 19, 2017
Feigin E., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2016. No. arXiv:1606.05219.
We introduce the notion of generalized Weyl modules for twisted current algebras. We study their representation-theoretic and combinatorial properties and connection to the theory of nonsymmetric Macdonald polynomials. As an application we compute the dimension of the classical Weyl modules in the remaining unknown case. ...
Added: June 17, 2016
Feigin E., Kato S., Makedonskyi I., Journal fuer die reine und angewandte Mathematik 2020 Vol. 764 P. 181-216
We study the non-symmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the non-symmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on ...
Added: August 12, 2020
Feigin E., Makedonskyi I., Mathematical Research Letters 2017 Vol. 24 No. 3 P. 741-766
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 $\mosp(1,2)$ 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 $t=0$ and $t=\infty$ specializations
of ...
Added: September 2, 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
Feigin E., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2015. No. 1512.03254.
The classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the ...
Added: December 15, 2015
Feigin E., Makedonskyi I., Journal of Combinatorial Theory, Series A 2015 P. 60-84
The Cherednik–Orr conjecture expresses the t →∞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: May 20, 2015
Olshanski G., Communications in Mathematical Physics 2021 Vol. 385 P. 595-631
We introduce and study a family of (q, t)-deformed discrete N-particle beta ensembles, where q and t are the parameters of Macdonald polynomials. The main result is the existence of a large-N limit transition leading to random point processes with infinitely many particles. ...
Added: June 22, 2021
Cherednik I., Feigin E., Advances in Mathematics 2015 Vol. 282 P. 220-264
Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t→∞ and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the ...
Added: September 3, 2015
Olshanski G., Working papers by Cornell University. Series math "arxiv.org" 2020
Using Okounkov's q-integral representation of Macdonald polynomials we construct an infinite sequence Ω1,Ω2,Ω3,… of countable sets linked by transition probabilities from ΩN to ΩN−1 for each N=2,3,…. The elements of the sets ΩN are the vertices of the extended Gelfand-Tsetlin graph, and the transition probabilities depend on the two Macdonald parameters, q and t. These ...
Added: January 19, 2021
Finkelberg M., Braverman A., Shiraishi J., Providence : American Mathematical Society, 2014
Let G be an almost simple simply connected complex Lie group, and let G/U be its base affine space. In this paper we formulate a conjecture which provides a new geometric interpretation of the Macdonald polynomials associated to G via perverse coherent sheaves on the scheme of formal arcs in the affinizationof G/U. We prove ...
Added: March 5, 2015
Loktev S., Kato S., / Cornell University. Series arXiv "math". 2017. No. 1712.03508.
We construct a filtration on integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each Weyl module there is given by a corresponding level-restricted Kostka polynomial. This leads to an interpretation of level-restricted Kostka polynomials as ...
Added: December 11, 2017
Makedonskyi I., Feigin E., Теоретическая и математическая физика 2017 Т. 192 № 2 С. 284-306
Вводится понятие обобщенного модуля Вейля для скрученных алгебр токов. Изучаются теоретико-представленческие и комбинаторные свойства этих модулей, а также их связь с несимметрическими полиномами Макдональда. В качестве приложения вычисляется размерность классических модулей Вейля в случае, до сих пор остававшемся неизвестным. ...
Added: August 6, 2017
Finkelberg M. V., MATHEMATICAL SCIENCES 2013 Vol. 51 No. 596 P. 46-51
This is a survey of the author's and his collaboratots' recent works on the quasiflags' moduli spaces introduced by Gerard Laumon some 25 years ago. These spaces are used in the study of geometric Eisenstein series, quantum cohomology and K-theory of the flag varieties, Weyl modules, Nekrasov partition function of N=2 supersymmetric gauge quantum field ...
Added: February 14, 2013
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
Khoroshkin A., / arXiv.org. Series 1312.7053 "1312". 2013. No. 1312.7053.
The aim of this paper is to introduce the categorical setup which helps us to relate the theory of Macdonald polynomials and the theory of Weyl modules for current Lie algebras discovered by V.\,Chari and collaborators. We identify Macdonald pairing with the homological pairing on the ring of characters of the Lie algebra of currents. ...
Added: February 14, 2014
Belavin A. A., Bershtein M., Tarnopolsky G. M., 2013 No. 3 P. 1-36
A bstract We continue our study of the AGT correspondence between instanton counting on
${{{{{\ mathbb {C}}^ 2}}}\ left/{{{{\ mathbb {Z}} _p}}}\ right.} $ and Conformal field theories with
the symmetry algebra $\ mathcal {A}\ left ({r, p}\ right) $. In the cases r= 1, p= 2 and r= 2, p= 2
this algebra specialized to: $\ mathcal ...
Added: September 9, 2014
Окубо Ю. undefined., Journal of Physics: Conference Series 2017 Vol. 804 No. 012036 P. 1-8
We investigate the existence and the orthogonality of the generalized Jack symmetric functions which play an important role in the AGT relations. We show their orthogonality by deforming them to the generalized Macdonald symmetric functions. ...
Added: October 26, 2017
Olshanski G., Selecta Mathematica, New Series 2021 Vol. 27 Article 41
Using Okounkov’s q-integral representation of Macdonald polynomials we construct an infinite sequence Ω1,Ω2,Ω3,… of countable sets linked by transition probabilities from Ω𝑁 to Ω𝑁−1 for each 𝑁=2,3,…. The elements of the sets Ω𝑁 are the vertices of the extended Gelfand–Tsetlin graph, and the transition probabilities depend on the two Macdonald parameters, q and t. These ...
Added: June 4, 2021