Papers by Arkady Berenstein

If you have problems downloading any of these PDF files please email me at: arkadiy@uoregon.edu


 

Symplectic groups over noncommutative algebras (with D. Alessandrini, V. RetakhE. RogozinnikovA. Wienhard)

We introduce the symplectic group Sp2(A,σ) over a noncommutative algebra A with an anti-involution σ. We realize several classical Lie groups as Sp2 over various noncommutative algebras, which provides new insights into their structure theory.We construct several geometric spaces, on which the groups Sp2(A,σ) act. We introduce the space of isotropic A-lines, which generalizes the projective line. We describe the action of Sp2(A,σ) on isotropic A-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic A-lines as invariants of this action. When the algebra A is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space XSp2(A,σ), and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as Sp2(A,σ)) and their complexifications, we obtain different models of the symmetric space as noncommutative generalizations of models of the hyperbolic plane and of the three-dimensional hyperbolic space. We also provide a partial classification of Hermitian algebras in Appendix A.

 

Geometric multiplicities (with Y. Li), preprint

In this paper, we introduce geometric multiplicities, which are positive varieties with potential fibered over the Cartan subgroup H of a reductive group G. They form a monoidal category and we construct a monoidal functor from this category to the representations of the Langlands dual group Gv of G. Using this, we explicitly compute various multiplicities in Gv-modules in many ways. In particular, we recover the formulas for tensor product multiplicities of Berenstein-Zelevinsky and generalize them in several directions. In the case when our geometric multiplicity X is a monoid, i.e., the corresponding Gv-module is an algebra, we expect that in many cases, the spectrum of this algebra is an affine Gv-variety Xv, and thus the correspondence X-->Xv has a flavor of both the Langlands duality and mirror symmetry.
 

Langlands Duality and Poisson-Lie Duality via Cluster Theory and Tropicalization (with A. Alekseev, B. Hoffman, Y. Li) Selecta Mathematica, 27 (4) DOI: 10.1007/s00029-021-00682-x.

Let G be a connected semisimple Lie group. There are two natural duality constructions that assign to it the Langlands dual group Gv and the Poisson-Lie dual group G*. The main result of this paper is the following relation between these two objects: the integral cone defined by the cluster structure and the Berenstein-Kazhdan potential on the double Bruhat cell Gv;wo,e. is isomorphic to the integral Bohr-Sommerfeld cone defined by the Poisson structure on the partial tropicalization of K* G* (the Poisson-Lie dual of the compact form K G). The first cone parametrizes the canonical bases of irreducible G-modules. The corresponding points in the second cone belong to integral symplectic leaves of the partial tropicalization labeled by the highest weight of the representation. As a by-product of our construction, we show that symplectic volumes of generic symplectic leaves in the partial tropicalization of K* are equal to symplectic volumes of the corresponding coadjoint orbits in k*. To achieve these goals, we use (Langlands dual) double cluster varieties defined by Fock and Goncharov. These are pairs of cluster varieties whose seed matrices are transpose to each other. There is a naturally defined isomorphism between their tropicalizations. The isomorphism between the cones described above is a particular instance of such an isomorphism associated to the double Bruhat cells G wo,e G and Gv;wo,e.

On cacti and crystals (with J. Greenstein and Jian-Rong Li) Representations and Nilpotent Orbits of Lie Algebraic Systems: in honor of the 75th Birthday of Tony Joseph,  Progress in Mathematics,   330, 2019.

In the present work we study actions of various groups generated by involutions on the category Oqint(g) of integrable highest weight Uq(g)-modules and their crystal bases for any symmetrizable Kac-Moody algebra g. The most notable of them are the cactus group and (yet conjectural) Weyl group action on any highest weight integrable module and its lower and upper crystal bases. Surprisingly, some generators of cactus groups are anti-involutions of the Gelfand-Kirillov model for Oqint(g) closely related to the remarkable quantum twists discovered by Kimura and Oya.

Poisson structures and potentials (with A. Alekseev, B. Hoffman, Y. Li) Lie Groups, Geometry, and Representation Theory: A Tribute to the Life and Work of Bertram Kostant, Birkhauser, 2018.

We introduce a notion of weakly log-canonical Poisson structures on positive varieties with potentials. Such a Poisson structure is log-canonical up to terms dominated by the potential. To a compatible real form of a weakly log-canonical Poisson variety we assign an integrable system on the product of a certain real convex polyhedral cone (the tropicalization of the variety) and a compact torus. We apply this theory to the dual Poisson-Lie group G* of a simply-connected semisimple complex Lie group G. We define a positive structure and potential on G* and show that the natural Poisson-Lie structure on G* is weakly log-canonical with respect to this positive structureand potential. For KG the compact real form, we show that the real form K*G* is compatible and prove that the corresponding integrable system is defined on the product of the decorated string cone and the compact torus of dimension 1/2(dim G - rank G).

Noncommutative Catalan numbers (with V. Retakh) Annals of Combinatorics, Vol. 23, Issue 3–4 (2019), 527–547.

The goal of this paper is to introduce and study noncommutative Catalan numbers Cn which belong to the free Laurent polynomial algebra in n generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman (q,t)-versions, another -- to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices Hn and introduce accompanying noncommutative binomial coefficients.

Factorizable module algebras (with K. Schmidt)  Int. Math. Res. Not. 2019 (21), 6711–6764 (2019).

The aim of this paper is to introduce and study a large class of g-module algebras which we call factorizable by generalizing the Gauss factorization of (square or rectangular) matrices. This class includes coordinate algebras of corresponding reductive groups G, their parabolic subgroups, basic affine spaces and many others. It turns out that tensor products of factorizable algebras are also factorizable and it is easy to create a factorizable algebra out of virtually any g-module algebra. We also have quantum versions of all these constructions in the category of Uq(g)-module algebras. Quite surprisingly, our quantum factorizable algebras are naturally acted on by the quantized enveloping algebra Uq(g*) of the dual Lie bialgebra g* of g.

Hecke-Hopf algebras (with D. Kazhdan)  Advances in Mathematics, Vol. 353 (2019), 312–395.
Let W be a Coxeter group. The goal of the paper is to construct new Hopf algebras contain Hecke algebras
Hq(W) as (left) coideal subalgebras. Our Hecke-Hopf algebras  H(W) have a number of applications. In particular they provide new solutions of quantum Yang-Baxter equation and lead to a construction of a new family of endo-functors of the category of Hq(W)-modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras; for an arbitrary Coxeter group W  the “Demazure” part of H(W) is being acted upon by generalized braided derivatives which generate the corresponding (generalized) Nichols algebra.
 

Canonical bases of quantum Schubert cells and their symmetries (with J. Greenstein),   J. Sel. Math. New Ser., doi:10.1007/s00029-017-0316-8.
The goal of this work is to provide an elementary construction of the canonical basis B(w) in each quantum Schubert cell
Uq(w) and to establish its invariance under modified Lusztig’s symmetries. To that effect, we obtain a direct characterization of the upper global basis Bup in terms of a suitable bilinear form and show that B(w) is contained in Bup and its large part is preserved by modified Lusztig’s symmetries.


Noncommutative marked surfaces (with V. Retakh), Advances in Mathematics,
Vol 328 (2018), 1010–1087.

The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface Σ. This is a noncommutative algebra AΣ generated by “noncommutative geodesics” between marked points subject to certain triangular relations and noncommutative analogues of Ptolemy-Plucker relations. It turns out that the algebra AΣ exhibits a noncommutative Laurent Phenomenon with respect to any triangulation of Σ, which confirms its “cluster nature.” As a surprising byproduct, we obtain a new topological invariant of Σ, which is a free or a 1-relator group easily computable in terms of any triangulation of Σ.   Another application is the proof of Laurentness and positivity of certain discrete noncommutative integrable systems.

 

Generalized adjoint actions (with V. Retakh), Journal of Lie Theory, 26 (2016), No. 1, 219225.

The aim of this paper is to generalize the classical formula exye-xk≥0 1/k! (ad x)k(y). We also obtain combinatorial applications to q-exponentials, q-binomials, and Hall-Littlewood polynomials.

Generalized Joseph’s decompositions
(with J. Greenstein), Comptes Rendus Mathematique, Doi : 10.1016/j.crma.2015.07.002.
We generalize the decomposition of Uq(g) introduced by A. Joseph and relate it, for g semisimple, to the celebrated computation of central elements due to V. Drinfeld. In that case we construct a natural basis in the center of Uq(g) whose elements behave as Schur polynomials and thus explicitly identify the center with the ring of symmetric functions.


Integrable clusters
(with J. Greenstein, D. Kazhdan), Comptes Rendus Mathematique,Vol 353, 5 (2015), 387–390.
The goal of this note is to study quantum clusters in which cluster variables (not coefficients) commute which each other. It turns out that this property is preserved by mutations. Remarkably, this is equivalent to the celebrated sign coherence conjecture recently proved by M. Gross, P. Hacking, S. Keel and M. Kontsevich.

 

Double canonical bases (with J. Greenstein), Advances in Mathematics,Vol. 316 (2017), 54111.

We introduce a new class of bases for quantized universal enveloping algebras Uq(g) and other doubles attached to semisimple and Kac-Moody Lie algebras. These bases contain dual canonical bases of upper and lower halves of Uq(g) and are invariant under many symmetries including all Lusztig’s symmetries if g is semisimple. It also turns out that a part of a double canonical basis of Uq(g) spans its center.

 

Mystic reflection groups (with Y. Bazlov), SIGMA 10 (2014), 040, 11 pages.

This paper aims to systematically study mystic reflection groups that emerged independently in a paper by the authors and in a paper by Kirkman, Kuzmanovich, and Zhang. A detailed analysis of this class of groups reveals that they are in a nontrivial correspondence with complex reflection groups G(m,p,n). We also prove that the group algebras of corresponding groups are isomorphic and classify all such groups up to isomorphism.

 

Quantum cluster characters of Hall algebras (with D. Rupel), Selecta Mathematica, DOI: 10.1007/s00029-014-0177-3.

The aim of the paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field Fq and any sequence i of simple objects in C the element XV,i of the corresponding algebra PC,i of q-polynomials. We prove that if C was hereditary, then the assignments VàXV,i define algebra homomorphisms from the (dual) Hall-Ringel algebra of C to the PC,i, which generalize the well-known Feigin homomorphisms from the upper half of a quantum group to q-polynomial algebras. If C is the representation category of an acyclic valued quiver (Q,d) and i=(io,io), where io 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 [29] and [30]. 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 [5] 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 [6].

 

Cocycle twists and extenstions of braided doubles (with Y. Bazlov), Contemp. Math., 592 (2013), 19–70.

It is well known that central extensions of a group G correspond to 2-cocycles on G. Cocycles can be used to construct extensions of G-graded algebras via a version of the Drinfeld twist introduced by Majid. We show how to define the second cohomology group of an abstract monoidal category C, generalising the Schur multiplier of a finite group and the lazy cohomology of a Hopf algebra, recently studied by Schauenburg, Bichon, Carnovale and others. A braiding on C leads to analogues of Nichols algebras in C, and we explain how the recent work on twists of Nichols algebras by Andruskiewitsch, Fantino, Garcia and Vendramin fits in our context. In the second part of the paper we propose an approach to twisting the multiplication in braided doubles, which are a class of algebras with triangular decomposition over G. Braided doubles are not G-graded, but may be embedded in a double of a Nichols algebra, where a twist is carried out. This is a source of new algebras with triangular decomposition. As an example, we show how to twist the rational Cherednik algebra of the symmetric group by the cocycle arising from the Schur covering group, obtaining the spin Cherednik algebra introduced by Wang.

 

Macdonald Polynomials and BGG reciprocity for current algebras (with M. Bennett, V. Chari, A. Khoroshkin, S. Loktev), Selecta Mathematica,  Vol. 20,  2 (2014),  585–607.

We study the category of graded representations with finite-dimensional graded pieces for the current algebra associated to a simple Lie algebra. This category has many similarities with the category O of modules for g and in this paper, we use the combinatorics of Macdonald polynomials to prove an analogue of the famous BGG duality in the case of sln+1.


Primitively generated Hall algebras
(with J. Greenstein), Pacific Journal of Mathematics, Vol. 281, No. 2, 2016.
The aim of the present paper is to demonstrate that Hall algebras of a large class of finitary exact categories behave like quantum nilpotent groups in the sense that they are generated by their primitive elements. Another goal is to construct analogues of quantum enveloping algebras as certain primitively generated subalgebras of the Hall algebras and conjecture an analogue of “Lie correspondence” for those finitary categories.

Triangular bases in quantum cluster algebras (with A. Zelevinsky), Int. Math. Res. Not. 2012, no. 21, 4821–4883.

A lot of recent activity has been directed towards various constructions of “natural” bases in cluster algebras. We develop a new approach to this problem which is close in spirit to Lusztig’s construction of a canonical basis, and the pioneering construction of the Kazhdan-Lusztig basis in a Hecke algebra. The key ingredient of our approach is a new version of Lusztig’s Lemma that we apply to all acyclic quantum cluster algebras. As a result, we construct the “canonical” basis in every such algebra that we call the canonical triangular basis.

The reciprocal of Σn≥0­­ a­­­­n­bn­ for non-commuting a and b, Catalan numbers and non-commutative quadratic equations (with  V. Retakh, C. Reutenauer, D. Zeilberger ), Contemp. Math. 592 (2013), 103–109.

The aim of this paper is to describe the inversion of the sum Σn≥0­­ a­­­­n­bn where a and b are non-commuting variables as a formal series in a and b. We show that the inversion satisfies a non-commutative quadratic equation and that the number of certain monomials in its homogeneous components equals to a Catalan number. We also study general solutions of similar quadratic equations.

Quantum Chevalley groups (with J. Greenstein), Contemp. Math., 592 (2013), 71–102.
The goal of this paper is to construct quantum analogues of Chevalley groups inside completions of quantum groups or, more precisely, inside completions of Hall algebras of finitary categories. In particular, we obtain pentagonal and other identities in the quantum Chevalley groups which generalize their classical counterparts and explain Faddeev-Volkov quantum dilogarithmic identities and their recent generalizations due to Keller.

Littlewood-Richardson coefficients for reflection groups
 (with E. Richmond), Advances in Mathematics, Vol 284 (2015), 54–111.

In this paper we explicitly compute all Littlewood-Richardson coefficients for semisimple or Kac-Moody groups G, that is, the structure coefficients of the cohomology algebra  H*(G/P), where P is a parabolic subgroup of G. These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood-Richardson coefficients is purely combinatorial and is given in terms of the Cartan matrix and the Weyl group of G. In particular, our formula gives a combinatorial proof of positivity of the Littlewood-Richardson coefficients in the cases when off-diagonal Cartan matrix entries are less than or equal to -2. Moreover, all our results for the Littlewood-Richardson coefficients extend to the structure coefficients of the T-equivariant cohomology algebra H*T(G/P).

A short proof of Kontsevich cluster conjecture (with V. Retakh), C. R. Math.
Acad. Sci., Paris 349 (2011), no. 3-4, 119–122.
We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map Kr:(x,y)→(xyx-1,(1+y-r)x-1) are given by noncommutative Laurent polynomials.

Stability inequalities and universal Schubert calculus of rank 2 (with M. Kapovich), Transformation Groups, Vol.
  16, Issue 4 (2011),  955-1007.
The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group W. That is, to each “sufficiently rich” spherical building Y of type W we associate a certain cohomology theory and verify that, first, it depends only on W (i.e., all such buildings are “homotopy equivalent”) and second, the cohomology ring is the associated graded of the coinvariant algebra of W under certain filtration. We also construct the dual homology “pre-ring” of Y. The convex “stability” cones defined via these (co)homology theories of Y are then shown to solve the problem of classifying weighted semistable m-tuples on Y in the sense of Kapovich, Leeb and Millson equivalently, they are cut out by the generalized triangle inequalities for thick Euclidean buildings with the Tits boundary Y. Quite remarkably, the cohomology ring is obtained from a certain universal algebra A by a kind of “crystal limit” that has been previously introduced by Belkale-Kumar for the cohomology of flag varieties and Grassmannians. Another degeneration of A leads to the homology theory of Y.

Quantum folding (with J. Greenstein)Int. Math. Res. Not. 2011, no. 21, 4821-4883.
In the present paper we introduce a quantum analogue of the classical folding of a simply-laced Lie algebra g to the non-simply-laced algebra gσ along a Dynkin diagram automorphism σ of g. For each quantum folding we replace gσ by its Langlands dual (gσ)v and construct a nilpotent Lie algebra n which interpolates between the nilpotent parts of g and (gσ)v, together with its quantized enveloping algebra Uq(n) and a Poisson structure on S(n). Remarkably, for the pair (g, (gσ)v)=(so2n+2,sp2n), the algebra Uq
(n) admits an action of the Artin braid group Brn and contains a new algebra of quantum n x n matrices with an adjoint action of Uq(sln), which generalizes the algebras constructed by K. Goodearl and M. Yakimov. The hardest case of quantum folding is, quite expectably, the pair (so8,G2) for which the PBW presentation of Uq(n) and the corresponding Poisson bracket on S(n) contain more than 700 terms each.

Quasiharmonic polynomials for Coxeter groups and representations of Cherednik algebras (with Yu. Burman),  Trans. Amer. Math. Soc., 362 (2010), 229–260.
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.

Dunkl Operators and Canonical Invariants of Reflection Groups
(with Yu. Burman), SIGMA 5 (2009), 057, 18 pages.

Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.

Affine buildings for dihedral groups  (with M. Kapovich),  Geometriae Dedicata, 156 (2012), 171-207.
We construct rank 2 thick nondiscrete affine buildings associated with an arbitrary finite dihedral group.


Noncommutative Dunkl operators and braided Cherednik algebras (with Y. Bazlov)  Selecta Mathematica14, (2009), 325–372.
We introduce braided Dunkl operators that are acting on a q-polynomial algebra and q-commute. Generalizing the approach of Etingof and Ginzburg, we explain the q-commutation phenomenon by constructing braided Cherednik algebras for which the above operators form a representation. We classify all braided Cherednik algebras using the theory of braided doubles developed in our previous paper. Besides ordinary rational Cherednik algebras, our classification gives new algebras attached to an infinite family of subgroups of even elements in complex reflection groups, so that the corresponding braided Dunkl operators pairwise anti-commute. We explicitly compute these new operators in terms of braided partial derivatives and divided differences.

Braided Doubles and rational Cherednik algebras (with Y. Bazlov), Advances in Mathematics, Vol. 220 (2009) 5,  1466–1530.
We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasi-Yetter-Drinfeld (QYD) modules over Hopf algebras which turn out to be a generalisation of the ordinary Yetter-Drinfeld modules. To each braiding (a solution to the braid equation) we associate a QYD-module and the corresponding braided Heisenberg double — this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by Nichols-Woronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds into the braided Heisenberg double attached to the corresponding complex reflection group.

Lie algebras and Lie groups over noncommutative rings (with V. Retakh), Advances in Mathematics, Vol. 218, 6, (2008), 1723–1758.
The aim of this paper is to introduce and study Lie algebras over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the F-current Lie algebra (g,A)(F), which is the Lie subalgebra of F\otimes A generated by F\otimes g. In most examples A is the universal enveloping algebra of g. Our description of the current algebra has a striking resemblance to the commutator expansions of F used by M. Kapranov in his approach to noncommutative geometry. We also associate with each Lie algebra (g,A)(F) a “noncommutative algebraic” group G which naturally acts on (g,A)(F) by conjugations and conclude the paper with a number of examples of such groups.

Braided 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 algebras with their classical counterparts.


Lecture notes on geometric crystals and their combinatorial analogues (with D. Kazhdan), Combinatorial aspect of integrable systems, MSJ Memoirs, 17, Mathematical Society of Japan,  Tokyo, 2007.
This is an exposition of the results on Geometric crystals and the associated Kashiwara crystal bases (presented by the first author in RIMS, August 2004).

Geometric and Unipotent Crystals II: From Unipotent Bicrystals to Crystal Bases (with D. Kazhdan), Contemp. Math., 433, Amer. Math. Soc., Providence, RI, 2007, 13–88.
For each reductive algebraic group G, we introduce and study unipotent bicrystals which serve as a regular version of rational 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.

 

Noncommutative 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 coefficients.

 

Quantum cluster 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.

 

Cluster algebras III: Upper bounds and double Bruhat cells (with S. Fomin and A. Zelevinsky)  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 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.

 

Tensor product 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.”

 

Geometric and 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
àT and a compatible family ei:Gm×XàX, 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. Surprisingly 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 isomorphism Xà>γ-1(eT 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 geometric crystal.

 

Coadjoint orbits, 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) Comment. 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.

 

Parametrizations 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.

 

Group-like elements in quantum groups and Feigin’s conjecture, preprint.
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 Ar and piecewise-linear combinatorics (with A. Zelevinsky), Duke 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 Ar (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 quantum 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 [8] that B contains all products of pairwise q-commuting elements of B. The conjecture was  proved in [8] 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).

Groups generated by involutions, Gel’fand-Tsetlin patterns, and combinatorics of Young tableaux (with Anatol Kirillov), Algebra 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 [7]; they involve the famous Schützenberger involution which also belongs to Gn.  Another result of [7] is a conjectural description of Kashiwara’s crystal operators for type A, in terms of Gn.

 

Triple multiplicities for sl(r+1) and the spectrum of the exterior algebra of the adjoint representation (with A. Zelevinsky), J. Algebraic Combin. 1 (1992), no. 1, 7–22.

 

When is the weight multiplicity equal to 1  (Russian) (with  A. Zelevinsky) Funkc. Anal. Pril. 24 (1990), no. 4, 1–13; translation: Funct. Anal. Appl. 24 (1990), no. 4, 259–269.

Tensor product multiplicities and convex polytopes in partition space (with A. Zelevinsky) J.

 Geom. 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.

Involutions on Gelfand-Tsetlin patterns and multiplicities in skew GL(n)-modules (with A. Zelevinsky) Soviet Math. Dokl. 37 (1988), no. 3, 799–802 592 (2013), 71–102.