Geometric
and
Unipotent Crystals II: From Unipotent Bicrystals to Crystal Bases
(with
D. Kazhdan)
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 G^\vee-modules, where G^\vee
is the Langlands dual groups. In fact, our analogues of crystal
bases
(which we refer to as crystals associated to G^\vee-modules)
are associated to G^\vee-modules directly, i.e., without
quantum
deformations.
Quasiharmonic
polynomials for Coxeter groups and representations of Cherednik
algebras
(with
Yu.
Burman)
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)
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.
Noncommutative
Double Bruhat cells and their factorizations (with
V.
Retakh)
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)
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)
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.
Geometric
and
unipotent crystals (with
D.
Kazhdan)
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
gamma
:X-->T and a compatible family e_i:G_m
times X-->X,
i
in
I
of rational actions of the multiplicative group G_m
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 G^ in the recent work by A. Braverman and D.
Kazhdan
come from geometric crystals. There are many examples of
positive
geometric crystals on (G_m)^l, i.e., those
geometric
crystals for which the actions e_i 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 G^. An emergence of G^
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--> gamma^-1(e) timesT for any
geometric
SL_n-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.
Tensor
product
multiplicities, canonical bases and totally positive varieties
(with A.
Zelevinsky)
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."
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 Delta_0. This is a rational polyhedral cone spanned by
all those dominant G-weights lambda for which the
simple
G-module
V_lambda
contains a non-trivial H-invariant. Our result generalizes the
result
by Klyachko who has solved this problem for G=GL_ntimes GL_ntimesGL_n
with the subgroup
H=GL_n embedded diagonally into G.
We describe the facets of the cone Delta_0 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 Delta,
which is spanned by those pairs (lambda,lambda') for
which
the restriction to H of the simple G-module
V_lambda
contains a simple H-module
V'_\lambda'.
Concavity of
weighted
arithmetic means with applications (with
Alex
Vainshtein),
Arch.
Math. (1997) 69, 120--126.
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,
55-66.
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 , J.
Algebra,to appear.
In this paper analogue of the Gelfand-Kirillov conjecture for any
simple
quantum group G_q is proved (here G_q is the q-deformed
coordinate ring of a simple algebraic group G). Namely, the
field
of fractions of G_q 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 G_q (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 w_0 in S_{r+1}
on the
dual canonical basis in each simple sl_{r+1}-module. Having
been
translated into the language of Gelfand-Tsetlin patterns and
Young
tableaux, this involution coincides with the Schutzenberger 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 A_r (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 A_2 and A_3.
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 S_n on Young tableaux,
discovered
by Lascoux and Schutzenberger. We introduced an action of S_n
by
piecewise-linear transformations on the space of Gelfand-Tsetlin
patterns.
In our approach, this group appears as a subgroup of the infinite group
G_n,
generated by quite simple piecewise-linear involutions (these
involutions
are continuous analogues of Bender-Knuth involutions acting on Young
tableaux).
The structure of G_n is not yet completely understood. Some
relations
were given in [7]; they involve the famous Sch\"utzenberger involution
which also belongs
to G_n. Another result of [7] is a conjectural
description
of Kashiwara's crystal operators for type A, in terms of G_n.
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, Vsesoyuz. Nauch.-Issled. Inst. Sistem. Issled., 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.