If C is the representation category of an acyclic valued quiver (Q,d) and i=(i0,i0), where i0 is a repetition-free source-adapted sequence, then we prove that the i-character XV,i equals the quantum cluster character XV introduced earlier by the second author in  and . Using this identification, we deduce a quantum cluster structure on the quantum unipotent cell corresponding to the square of a Coxeter element. As a corollary, we prove a conjecture from the joint paper  of the first author with A. Zelevinsky for such quantum unipotent cells. As a byproduct, we construct the quantum twist and prove that it preserves the triangular basis introduced by A. Zelevinsky and the first author in .Cocycle twists and extenstions of braided doubles (with Y. Bazlov) Contemp. Math. 592 (2013), 19-70.
polynomials for Coxeter groups and representations of Cherednik algebras (with
Trans. Amer. Math. Soc., 362
We introduce and study deformations of finite-dimensional modules over rational Cherednik algebras. Our main tool is a generalization of usual harmonic polynomials for Coxeter groups – the so-called quasiharmonic polynomials. A surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all Coxeter groups.
symmetric and exterior algebras (with S. Zwicknagl)
Trans. Amer. Math. Soc., 360 (2008), 3429–3472.
We introduce and study symmetric and exterior algebras in braided monoidal categories such as the category O over quantum groups. We relate our braided symmetric algebras and braided exterior algebas with their classical counterparts.
Unipotent Crystals II: From Unipotent Bicrystals to Crystal Bases
(with D. Kazhdan)
Contemp. Math., 433, Amer. Math. Soc., Providence, RI,
For each reductive algebraic group G, we introduce and study unipotent bicrystals which serve as a regular version of birational geometric and unipotent crystals introduced earlier by the authors. The framework of unipotent bicrystals allows, on the one hand, to study systematically such varieties as Bruhat cells in G and their convolution products and, on the other hand, to give a new construction of many normal Kashiwara crystals including those for Gv-modules, where Gv is the Langlands dual groups. In fact, our analogues of crystal bases (which we refer to as crystals associated to Gv-modules) are associated to Gv-modules directly, i.e., without quantum deformations.
Double Bruhat cells and their factorizations (with V. Retakh) Int. Math. Res. Not., 8 (2005), 477–516.
In the present paper we study noncommutative double Bruhat cells. Our main results are explicit positive matrix factorizations in the cells via quasiminors of matrices with noncommutative coefficents.
algebras (with A. Zelevinsky)
Advances in Mathematics, vol. 195,
2 (2005), 405–455.
Cluster algebras were introduced by S. Fomin and A. Zelevinsky; their study continued in a series of papers including Cluster algebras III: Upper bounds and double Bruhat cells. This is a family of commutative rings designed to serve as an algebraic framework for the theory of total positivity and canonical bases in semisimple groups and their quantum analogs. In this paper we introduce and study quantum deformations of cluster algebras.
algebras III: Upper bounds and double Bruhat cells (with S. Fomin and
Duke Math. Journal, vol. 126,
1 (2005), 1–52.
We continue the study of cluster algebras. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon, we show that, under an assumption of "acyclicity", a cluster algebra coincides with its "upper" counterpart, and is finitely generated. In this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to the upper cluster algebra explicitly defined in terms of relevant combinatorial data.
multiplicities, canonical bases and totally positive varieties (with
A. Zelevinsky) Invent. Math., vol. 143, 1 (2001), 77–128.
We obtain a family of explicit ``polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. Here ``polyhedral" means that the multiplicity in question is expressed as the number of lattice points in some convex polytope. Our answers use a new combinatorial concept of i-trails which resemble Littelmann's paths but seem to be more tractable. We also study combinatorial structure of Lusztig's canonical bases or, equivalently of Kashiwara's global bases. Although Lusztig's and Kashiwara's approaches were shown by Lusztig to be equivalent to each other, they lead to different combinatorial parametrizations of the canonical bases. One of our main results is an explicit description of the relationship between these parametrizations. Our approach to the above problems is based on a remarkable observation by G. Lusztig that combinatorics of the canonical basis is closely related to geometry of the totally positive varieties. We formulate this relationship in terms of two mutually inverse transformations: ``tropicalization" and ``geometric lifting."
unipotent crystals (with D. Kazhdan) Geom. Funct. Anal., Special Volume,
Part I (2000), 188–236.
We introduce geometric crystals and unipotent crystals which are algebro-geometric analogues of Kashiwara's crystal bases. Given a reductive group G, let I be the set of vertices of the Dynkin diagram of G and T be the maximal torus of G. The structure of a geometric G-crystal on an algebraic variety X consists of a rational morphism γ:X–>T and a compatible family ei:Gm×X–>X, i in I of rational actions of the multiplicative group Gm satisfying certain braid-like relations. Such a structure induces a rational action of W on X. Surprizingly many interesting rational actions of the group W come from geometric crystals. Also all the known examples of the action of W which appear in the construction of Gamma-functions for the representations of the Langlands dual group Gv in the recent work by A. Braverman and D. Kazhdan come from geometric crystals. There are many examples of positive geometric crystals on (Gm)l, i.e., those geometric crystals for which the actions ei and the morphism gamma are given by positive rational expressions. One can associate to each positive geometric crystal X the Kashiwara's crystal corresponding to the Langlands dual group Gv. An emergence of Gv in the "crystal world" was observed earlier by G. Lusztig. Another application of geometric crystals is a construction of trivialization which is an W-equivariant isomorhism X-->γ-1(e)×T for any geometric SLn-crystal. Unipotent crystals are geometric analogues of normal Kashiwara crystals. They form a strict monoidal category. To any unipotent crystal built on a variety X we associate a certain gometric crystal.
moment polytopes, and the Hilbert-Mumford criterion (with R. Sjamaar), J. Amer. Math. Soc., 13 (2000),
no. 2, 433–466.
In this paper we solve of the following problem: Given a reductive group G, and its reductive subgroup H, describe the momentum cone Δo. This is a rational polyhedral cone spanned by all those dominant G-weights λ for which the simple G-module Vλ contains a non-trivial H-invariant. Our result generalizes the result by Klyachko who has solved this problem for G= GLn×GLn×GLn with the subgroup H=GLn embedded diagonally into G. We describe the facets of the cone Δo in terms of the ``relative'' Schubert calculus of the flag varieties of the two groups. Another formulation of the result is the description of the relative momentum cone Δ, which is spanned by those pairs (λ,λ') for which the restriction to H of the simple G-module Vλ contains a simple H-module V'λ'.
Domino tableaux, Schutzenberger involution and action of the symmetric group (with Anatol Kirillov), Proceedings of the 10th International Conference on Formal Power Series and Algebraic Combinatorics, Fields Institute, Toronto, 1998, Discrete Math., vol. 225, 1–3 (2000), 5–24.
Concavity of weighted arithmetic means with applications (with Alex Vainshtein), Arch. Math. (1997) 69, 120–126.
Total positivity in
Schubert varieties (with A. Zelevinsky)
Math. Helv. 72 (1997), no. 1, 128–166.
In this paper we further develop the remarkable parallelism discovered by Lusztig between the canonical basis and the variety of totally positive elements in the unipotent group.
of canonical bases and totally positive matrices (with S. Fomin and A. Zelevinsky),
Advances in Mathematics
122 (1996), 49–149.
We provide: (i) explicit formulas for Lusztig's transition maps related to the canonical basis of the quantum group of type A; (ii) formulas for the factorizations of a square matrix into elementary Jacobi matrices; (iii) a family of new total positivity criteria.
in quantum groups and Feigin's conjecture, to appear in J. Algebra.
In this paper analogue of the Gelfand-Kirillov conjecture for any simple quantum group Gq is proved (here Gq is the q-deformed coordinate ring of a simple algebraic group G). Namely, the field of fractions of Gq is isomorphic to the field of fractions of a certain skew-polynomial ring. The proof is based on a construction of some group-like elements in Gq (which are q-analogs of elements in G).
Canonical bases for the
quantum group of type A_r and piecewise-linear combinatorics
(with A. Zelevinsky)
Math. J. 82 (1996), no. 3, 473–502.
We use the structure theory of the dual canonical basis B is to obtain a direct representation-theoretic proof of the Littlewood-Richardson rule (or rather, its piecewise-linear versions discussed above).
Another application of string technique is an explicit formula for the action of the longest element wo in Sr+1 on the dual canonical basis in each simple slr+1-module. Having been translated into the language of Gelfand-Tsetlin patterns and Young tableaux, this involution coincides with the Schützenberger involution.
String bases for quantum
groups of type A_r (with A. Zelevinsky)
I. M. Gelfand Seminar, 51–89, Adv.
Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993.
We introduce and study a family of string bases for the guantum groups of type Ar (which includes the dual canonical basis). These bases are defined axiomatically and possess many interesting properties, e.g., they all are good in the sense of Gelfand and Zelevinsky. For every string basis, we construct a family of combinatorial labelings by strings. These labelings in a different context appeared in more recent works by M. Kashiwara and by P. Littelmann. We expect that B has a nice multiplicative structure. Namely, we conjecture in  that B contains all products of pairwise q-commuting elements of B. The conjecture was proved in  for A2 and A3. In fact, for r< 4, the dual canonical basis B is the only string basis and it consists of all q-commuting products of quantum minors (for r arbitrary, we proved that any string basis contains all quantum minors).
by involutions, Gel'fand-Tsetlin patterns, and combinatorics of Young tableaux
(with Anatol Kirillov),
i Analiz 7 (1995), no. 1, 92–152 (Russian). Translation in St. Petersburg
Math. J. 7 (1996), no. 1, 77-127
The original motivation of this paper was to understand a rather mysterious action of the symmetric group Sn on Young tableaux, discovered by Lascoux and Schutzenberger. We introduced an action of Sn by piecewise-linear transformations on the space of Gelfand-Tsetlin patterns. In our approach, this group appears as a subgroup of the infinite group Gn, generated by quite simple piecewise-linear involutions (these involutions are continuous analogues of Bender-Knuth involutions acting on Young tableaux). The structure of Gn is not yet completely understood. Some relations were given in ; they involve the famous Schützenberger involution which also belongs to Gn. Another result of  is a conjectural description of Kashiwara's crystal operators for type A, in terms of Gn.
for sl(r+1) and the spectrum of the exterior algebra of the adjoint
representation (with A.Zelevinsky)
Algebraic Combin. 1 (1992), no. 1, 7–22.
When is the weight multiplicity
equal to 1 (Russian) (with A. Zelevinsky)
Pril. 24 (1990), no. 4, 1–13; translation in Funct. Anal. Appl.
24 (1990), no. 4, 259–269.
Tensor product multiplicities
and convex polytopes in partition space(with A. Zelevinsky)
Phys. 5 (1988), no. 3, 453–472.
A multiplicative analogue of the Bergstrom inequality for a matrix product in the sense of Hadamard (with Alex Vainshtein) (Russian) Uspekhi Mat. Nauk 42 (1987), no. 6(258), 181–182. Translation: Russian Mathematical Surveys.
The convexity property of the Poisson distribution and its applications in queueing theory(with Alex Vainshtein and A. Kreinin) (Russian), Stability problems for stochastic models (Varna, 1985), 17–22, VINITI Moscow, 1986. Translation: J. Soviet Math. 47 (1989), no. 1.
Gelfand-Tsetlin patterns and multiplicities in skew GL(n)-modules(with
Dokl. 37 (1988), no. 3, 799–802.