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Generalized electrical Lie algebras (with A. Gainutdinov, V. Gorbunov),
preprint.
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
We count Gallai and transitive colorings of the
root system of type A using the maximal number of colors, and show that, when
equipped with a natural descent set map, the resulting quasisymmetric
function is symmetric and Schur-positive.
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), 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, 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), 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), 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,
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), 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)
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
Mathematica, 14, (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⊗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), 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: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,
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 Alek 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 Alek 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 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), 71–102.