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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 structureand 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, 2017).
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, 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), 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.
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\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: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.