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.
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 algebas with their classical
counterparts.
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 birational
geometric and unipotent crystals introduced earlier by the authors.
The framework of unipotent bicrystals allows, on the one hand, to study
systematically such varieties as Bruhat cells in G and their
convolution
products and, on the other hand, to give a new construction of many
normal
Kashiwara crystals including those for Gv-modules,
where Gv
is the Langlands dual groups. In fact, our analogues of crystal
bases
(which we refer to as crystals associated to Gv-modules)
are associated to Gv-modules
directly, i.e., without
quantum
deformations.
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 coefficents.
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 an upper cluster algebra, defined as an
intersection
of certain Laurent polynomial rings. Strengthening the Laurent
phenomenon,
we show that, under an assumption of "acyclicity", a cluster algebra
coincides
with its "upper" counterpart, and is finitely generated. In this case,
we also describe its defining ideal, and construct a standard monomial
basis. We prove that the coordinate ring of any double Bruhat cell in a
semisimple complex Lie group is naturally isomorphic to the upper
cluster
algebra explicitly defined in terms of relevant combinatorial data.
Tensor
product
multiplicities, canonical bases and totally positive varieties
(with A.
Zelevinsky)
Invent. Math., vol. 143, 1
(2001), 77–128.
We obtain a family of explicit ``polyhedral" combinatorial expressions
for multiplicities in the tensor product of two simple
finite-dimensional
modules over a complex semisimple Lie algebra. Here ``polyhedral" means
that the multiplicity in question is expressed as the number of lattice
points in some convex polytope. Our answers use a new combinatorial
concept
of i-trails which resemble Littelmann's paths but seem
to
be more tractable. We also study combinatorial structure of
Lusztig's
canonical bases or, equivalently of Kashiwara's global bases. Although
Lusztig's and Kashiwara's approaches were shown by Lusztig to be
equivalent
to each other, they lead to different combinatorial parametrizations of
the canonical bases. One of our main results is an explicit description
of the relationship between these parametrizations. Our approach to the
above problems is based on a remarkable observation by G. Lusztig that
combinatorics of the canonical basis is closely related to geometry of
the totally positive varieties. We formulate this relationship in terms
of two mutually inverse transformations: ``tropicalization" and
``geometric
lifting."
Geometric
and
unipotent crystals (with
D.
Kazhdan)
Geom. Funct. Anal., Special
Volume, Part I (2000), 188–236.
We introduce geometric crystals and unipotent crystals which are
algebro-geometric
analogues of Kashiwara's crystal bases. Given a reductive group G,
let I be the set of vertices of the Dynkin diagram of G
and
T
be the maximal torus of G. The structure of a geometric
G-crystal
on an algebraic variety X consists of a rational morphism
γ:X–>T
and a compatible family ei:Gm×X–>X,
i
in
I
of rational actions of the multiplicative group Gm
satisfying certain braid-like relations. Such a structure
induces
a rational action of W on X. Surprizingly many
interesting
rational actions of the group W come from geometric
crystals.
Also all the known examples of the action of W which
appear
in the construction of Gamma-functions for the representations of
the Langlands dual group Gv
in the recent work by A. Braverman and D.
Kazhdan
come from geometric crystals. There are many examples of
positive
geometric crystals on (Gm)l, i.e., those
geometric
crystals for which the actions ei and the morphism gamma
are given by positive rational expressions. One can associate to
each positive geometric crystal X the Kashiwara's crystal
corresponding to the Langlands dual group Gv.
An emergence of Gv
in the "crystal world" was observed earlier by G. Lusztig. Another
application
of geometric crystals is a construction of trivialization which
is an W-equivariant
isomorhism X-->γ-1(e)×T for any
geometric SLn-crystal.
Unipotent crystals are geometric analogues of normal Kashiwara
crystals.
They form a strict monoidal category. To any unipotent crystal built on
a variety X we associate a certain gometric crystal.
Coadjoint
orbits,
moment polytopes, and the Hilbert-Mumford criterion (with
R.
Sjamaar), J. Amer. Math. Soc.,
13 (2000), no. 2, 433–466.
In this paper we solve of the following problem: Given a reductive
group G, and its reductive subgroup H, describe the momentum
cone Δo.
This is a rational polyhedral cone spanned by
all those dominant G-weights λ for which the
simple
G-module Vλ
contains a non-trivial H-invariant. Our result generalizes the
result
by Klyachko who has solved this problem for G= GLn×GLn×GLn
with the subgroup
H=GLn
embedded diagonally into G.
We describe the facets of the cone Δo
in terms of the
``relative''
Schubert calculus of the flag varieties of the two groups. Another
formulation
of the result is the description of the relative momentum cone Δ,
which is spanned by those pairs (λ,λ') for
which
the restriction to H of the simple G-module Vλ contains a
simple H-module V'λ'.
Domino tableaux, Schutzenberger involution and action of the symmetric group (with Anatol Kirillov), Proceedings of the 10th International Conference on Formal Power Series and Algebraic Combinatorics, Fields Institute, Toronto, 1998, Discrete Math., vol. 225, 1–3 (2000), 5–24.
Concavity of weighted arithmetic means with applications (with Alex Vainshtein), Arch. Math. (1997) 69, 120–126.
Total
positivity in Schubert varieties (with A.
Zelevinsky)
Comment.
Math. Helv. 72 (1997), no. 1, 128–166.
In this paper we further develop the remarkable parallelism discovered
by Lusztig between the canonical basis and the variety of totally
positive
elements in the unipotent group.
Parametrizations
of canonical bases and totally positive matrices (with S.
Fomin and A.
Zelevinsky),
Advances
in Mathematics 122 (1996), 49–149.
We provide: (i) explicit formulas for Lusztig's transition maps related
to the canonical basis of the quantum group of type A; (ii) formulas
for
the factorizations of a square matrix into elementary Jacobi matrices;
(iii) a family of new total positivity criteria.
Group-like
elements
in quantum groups and Feigin's conjecture, to appear in J.
Algebra.
In this paper analogue of the Gelfand-Kirillov conjecture for any
simple
quantum group Gq
is proved (here Gq
is the q-deformed
coordinate ring of a simple algebraic group G). Namely, the
field
of fractions of Gq
is isomorphic to the field of fractions of
a
certain skew-polynomial ring. The proof is based on a construction of
some
group-like elements in Gq
(which are q-analogs of
elements
in G).
Canonical
bases for the quantum group of type A_r and piecewise-linear
combinatorics
(with
A.
Zelevinsky)
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 A_r
(with
A.
Zelevinsky)
I. M. Gelfand Seminar, 51–89, Adv.
Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI,
1993.
We introduce and study a family of string bases for the guantum
groups of type Ar
(which includes the dual canonical basis).
These
bases are defined axiomatically and possess many interesting
properties,
e.g., they all are good in the sense of Gelfand and
Zelevinsky.
For every string basis, we construct a family of combinatorial
labelings
by strings. These labelings in a different context appeared in
more
recent works by M. Kashiwara and by P. Littelmann. We expect that B
has a nice multiplicative structure. Namely, we conjecture in [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 in Funct.
Anal. Appl. 24 (1990), no. 4, 259–269.
Tensor
product multiplicities and convex polytopes in partition space(with
A.
Zelevinsky)
J.
Geom. Phys. 5 (1988), no. 3, 453–472.
A multiplicative analogue of the Bergstrom inequality for a matrix product in the sense of Hadamard (with Alex Vainshtein) (Russian) Uspekhi Mat. Nauk 42 (1987), no. 6(258), 181–182. Translation: Russian Mathematical Surveys.
The convexity property of the Poisson distribution and its applications in queueing theory(with Alex Vainshtein and A. Kreinin) (Russian), Stability problems for stochastic models (Varna, 1985), 17–22, VINITI Moscow, 1986. Translation: J. Soviet Math. 47 (1989), no. 1.
Involutions
on
Gelfand-Tsetlin patterns and multiplicities in skew
GL(n)-modules(with
A.
Zelevinsky)
Soviet
Math. Dokl. 37 (1988), no. 3, 799–802.