If you have problems downloading any of these PDF files please email me at: arkadiy@uoregon.edu
Hecke and Artin monoids and their homomorphisms (with J. Greenstein and J.-R. Li), preprint.
This work was motivated by a striking observation that parabolic projections of Hecke monoids map respect all parabolic elements. We found other classes of homomorphisms of Hecke monoids with the same property and discovered that many of them lift to homomorphisms of covering Artin monoids with a similar property. It turned out that they belong to a much larger class (in fact, a category) of homomorphisms of Artin monoids, most of which appear to be new.
Generalized
electrical Lie algebras (with A.
Gainutdinov, V. Gorbunov), submitted.
We generalize the electrical
Lie algebras originally introduced by Lam and Pylyavskyy in several ways. To
each Kac-Moody Lie algebra g we associate two types (vertex type and edge type) of the
generalized electrical algebras. The electrical Lie algebras of vertex type are
always subalgebras of g and are flat deformations of the nilpotent Lie subalgebra of g. In many cases including sln, son, and sp2n we find new (edge) models for
our generalized electrical Lie algebras of vertex type. Finding an edge model
in general is an interesting an open problem.
Valuations,
bijections, and bases (with D.
Grigoriev), preprint.
The aim of this paper is to build
a theory of commutative and noncommutative injective
valuations of various algebras. The targets of our valuations are
(well-)ordered commutative and noncommutative (partial or entire) semigroups
including any sub-semigroups of the free monoid Fn on n
generators and various quotients. In the case when the (partial) valuation
semigroup is finitely generated, we construct a generalization of the standard
monomial bases for the so-valued algebra, which seems to be new in
noncommutative case. Quite remarkably, for any pair of well-ordered valuations
one has canonical bijections between the valuation semigroups, which serve as
analogs of the celebrated Jordan-Hölder correspondences and these bijections
are “almost” homomorphisms of the involved (partial and entire) semigroups.
Transitive and Gallai
colorings (with R. M. Adin , J. Greenstein, J-R Li, A. Marmor, Y. Roichman), preprint.
Gallai coloring of
the complete graph is an edge-coloring with no rainbow triangle. This concept
first appeared in the study of comparability graphs and anti-Ramsey theory. We
introduce a transitive analogue for acyclic directed graphs, and generalize
both notions to Coxeter systems, matroids and
commutative algebras. It is shown that for any finite matroid (or oriented
matroid), the maximal number of colors is equal to the matroid rank. This
generalizes a result of Erdős-Simonovits-Sós for complete graphs. The
number of Gallai (or transitive) colorings of the matroid that use at most
Twists on rational Cherednik algebras (with Y. Bazlov, E. Jones-Healey, and A. McGaw), Quarterly Journal of Mathematics, 74 (2), 2022.
We show
that braided Cherednik algebras introduced by the first two authors are cocycle
twists of rational Cherednik algebras of the imprimitive complex reflection
groups
Symplectic groups over noncommutative algebras (with D. Alessandrini,
V. Retakh, E. Rogozinnikov, A. Wienhard)
Selecta Mathematica, 28,
82 (2022)
We introduce the symplectic
group
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 XàXv 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, 69 (2021)
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 J.-R. 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 structure and potential. For K⊂G 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), Selecta Mathematica, 23, .
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), pages 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, pages 219–225.
The aim of this paper is to generalize the
classical formula exye-x=Σk≥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),
pages 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), pages 54–111.
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, 21, pages 1121–1176 (2015).
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 extensions of braided
doubles (with Y.
Bazlov), Contemp.
Math., 592
(2013), pages 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), pages 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, pages 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), pages 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), pages 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, pages 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)
Stability inequalities and universal Schubert calculus of
rank 2 (with M. Kapovich), Transformation Groups, Vol. 16, Issue 4 (2011), pages
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, pages 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), pages 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 Mathematica, 14, (2009), pages 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, pages 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), pages 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⊗A generated
by F⊗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), pages 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, pages
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), pages 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), pages 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), pages 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), pages 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), pages 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:XàT 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(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 geometric crystal.
Coadjoint orbits, moment polytopes, and the
Hilbert-Mumford criterion (with R. Sjamaar), J. Amer. Math. Soc., 13 (2000),
no. 2, pages 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), pages 5–24.
Concavity of weighted arithmetic means with applications (with Alek Vainshtein), Arch. Math. (1997) 69, pages 120–126.
Total
positivity in Schubert varieties (with A. Zelevinsky) Comment. Math. Helv. 72 (1997), no. 1, pages 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),
pages 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, pages 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, pages 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, pages 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, pages 259–269.
Tensor product multiplicities and convex polytopes in partition space (with A. Zelevinsky) J. Geom. Phys. 5 (1988), no. 3, pages 453–472.
A multiplicative analogue of the Bergstrom inequality for a matrix product in the sense of Hadamard (with Alek Vainshtein) (Russian) Uspekhi Mat. Nauk 42 (1987), no. 6 (258), pages 181–182. Translation: Russian Mathematical Surveys.
The convexity property of the Poisson distribution and its applications in queueing theory (with Alek Vainshtein and A. Kreinin) (Russian). 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), pages 71–102.