Algebra and Combinatorics Seminar 

Fridays, Milner 317 3:003:50 PM 
Cayley Graph of the Free Product Z_{3} * Z_{5 } 

The Algebra and Combinatorics seminar is devoted to studying algebra, combinatorics, their interconnection, and their relations to mathematics and applications. The seminar's organizers are Frank Sottile and Luis Garcia.  

Abstract:
In 1980, Askold Khovanskii established his fewnomial bound for the number of
real solutions to a system of polynomials, thereby showing that the complexity
of real solutions to a polynomial system depends upon the number of monomials
and not the degree. This fundamental finiteness result in real algebraic
geometry was proven by induction on the number of monomials and the bound is
unrealistically large.
I will report on joint work with Frederic Bihan on a new fewnomial bound which
is substantially lower than Khovanskii's bound. This bound is obtained by
first reducing a given system to a Gale system, which comes from the Gale dual
to the exponent vectors in the original system, and then bounding the number of
solutions to a Gale system. Like Khovanskii's bound, this bound is the product
of an exponential function and a polynomial in the dimension, with the
exponents in both terms depending upon the number of monomials. In our bound,
the exponents are smaller than in Khovanskii's.
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Abstract:
Given an Ndimensional unitary representation of a discrete group G,
the closure of the image of G in U(N) is a compact group,
and one may
try to determine the closed image of G precisely. For braid groups,
there are several families of unitary representations known: for
example those factoring the TemperleyLieb, Hecke, and BirmanMurakamiWenzl algebras.
Although Jones explicitly asked this question in 1983 for the braid
group representations factoring over TemperleyLieb algebras, few
results were known until potential applications to quantum computing
inspired renewed interest around 2001.
I will discuss recent work with Larsen and Wang on the BirmanMurakamiWenzl algebra
cases, focusing on interesting subcases described in a recent
preprint
with Larsen. This talk should be
accessible to graduate students.
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Abstract:
Each of the traditional finite quantum groups is associated to a
Lie algebra and therefore has at its heart a Dynkin diagram or
related graph. A recent classification by Andruskiewitsch and
Schneider shows that the seemingly much larger class of finite
pointed Hopf algebras is not so different after all: Each such
Hopf algebra has at its heart a collection of linked Dynkin
diagrams from which its structure is largely determined.
In this talk we will give an overview of this classification. Time
permitting, we will also describe joint work with Mitja Mastnak
(Munich) in which we view the Hopf algebras of Andruskiewitsch
and Schneider from a different perspective, as deformations of
betterknown graded Hopf algebras. This perspective allows us to
use cohomology and deformation theory to understand these Hopf
algebras and potentially to fill in some gaps left in the
classification.
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Abstract:
There is a 11 correspondence between Schubert varieties in a
generalized flag variety G/P and the simple modules in a regular block of
the parabolic highest weight category O_P. The simple highest weight modules
that correspond to the (rationally) smooth Schubert varieties have many nice
properties that are typically associated with finite dimensional highest
weight modules. For example, these simple modules have a BGG resolution in terms
of generalized Verma modules.
In the first part of my talk, I will describe the above correspondence and
then present my recent work with Brian Boe on the classification of smooth
Schubert varieties in G/P for the special when G is of simplylaced type and P is a
maximal parabolic. In the second part of my talk, I will explain how this
all relates to my earlier work with Thomas Enright on the classical
determinantal varieties.
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Abstract:
Convex rank tests are partitions of the symmetric group corresponding to
polyhedral fans that coarsen the braid arrangement, and are equivalent to to
the probabilistic conditional independence structures known as
semigraphoids. Each equivalence class is the set of linear extensions of a
poset, a property that lends these objects to analyzing time series data
nonparametrically. Submodular rank tests are classified by the faces of the
cone of submodular functions, or by Minkowski summands of the permutohedron
(generalized permutohedra). Graphical tests correspond to both graphical
models and to graph associahedra. Connections to Markov bases and toric
geometry will also be explored.
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Abstract:
Let K be a field and let R be the polynomial ring in infinitely
many indeterminates over K. Since R is not Noetherian, computation in R,
and in particular, of Groebner bases, is impossible. However, suppose that
we consider ideals I which are invariant under the action of the infinite
symmetric group G. Such ideals correspond to R[G]submodules of R, and in
this setting, R is Noetherian and submodules I have (finite) Groebner
bases (as proved recently by Aschenbrenner and Hillar). We review this
story and describe a new result that gives an explicit method for
computing in R; in particular, solving the membership problem for
invariant ideals in R.
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Abstract:
We show how the number of real roots of certain nonlinear
systems of equations can be understood fairly completely in terms of
simple combinatorial diagrams. This is the gist of Viro's Theorem,
which we illustrate in terms of examples at the limits of the hypotheses
of this theorem.
Finding sparse polynomial systems with maximally many real roots
is an active research area, and we detail some of what is known (and not
known). We then conclude with a new extremal example, derived by the
speaker and two undergraduate REU students last July.
We assume no background in algebraic geometry.
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Abstract:
Eventhough the representation theory of the symmetric
group S_n over the field of complex numbers is well
understood, it is still unknown an efficient way to
compute the multiplicity of an irreducible character
of S_n in the Kronecker product of other two irreducible
characters of S_n. We call this multiplicities
Kronecker coefficients.
In this talk we reinterpret an old method for computing
Kronecker coefficients due to Littlewood in a purely
combinatorial way and use it to obtain some new results.
First we show that the multiplicity of an irreducible
character in the square of another irreducible character
can be computed by evaluating a polynomial with rational
coefficients in variables indexed by connected skew Young
diagrams.
Second we prove a new stability property for Kronecker
coefficients.
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Abstract:
In this talk, we give a new generalization of parking functions
called Gmultiparking functions. We show how this generalization is in
bijection to rooted spanning forests, and how a generalized
graphsearching algorithm can generate a family of such bijections. This
algorithm also yields a categorization of the edges of the underlying
graph. Time permitting, we will demonstrate how this categorization
provides a new presentation of the Tutte polynomial.
This is joint work with Catherine Yan.
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Abstract:
We call two permutations of the first n naturals colliding if
they map at least one number to consecutive naturals. We give bounds for
the exponential asymptotics of the largest cardinality of any set of
pairwise colliding permutations of [n]. We relate this problem to the
determination of the Shannon capacity of an infinite graph and initiate
the study of analogous problems for infinite graphs with finite chromatic
number. In collaboration with J\'anos K\"orner
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Abstract:
We give an introduction to the construction of
singular unitary representations of noncompact
semisimple groups. For exceptional groups, we
describe a number of methods to construct
a class of such representations, and investigate
a possible connection with exceptional versions of
Howe duality.
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Abstract:
In recent years, multiplier ideals have been used to answer
several questions in algebraic geometry. I will introduce multiplier
ideals, give some basic examples, and show some applications.
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Abstract:
In his 1890 paper on linear invariants of the special linear group, Hilbert
used two basic tools to prove the finite generation of the ring of
invariants.
One was his basis theorem that he proved for this purpose, and the other was
a differential operator defined on matrices called the Omega process, and
that was well known and extensively used previously to construct invariants.
Modern proofs of this finite generation have substituted the Omega process
by the complete reducibility of the representations, a result that was
unknown
to Hilbert and proved by Weyl in the 20s, using the so called unitarian
restriction.
In this talk we go back to considering Hilbert's method and propose a
general
definition of Omega process (only defined by Cayley for the variety of all
matrices) and following Hilbertwe show how to use the process in order
to prove the finite generion of invariants: The set up for this general
definition is the theory of algebraic monoids with zero.
Finally we show that an arbitrary reductive monoid admits an Omega process,
thus providing a proof of Weyl's theorem on the finite generation of the
invariants for an arbitrary affine reductive group.
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Abstract:
Let G be a nontrivial, loopless multigraph. For each
nontrivial subgraph H of G, let g(H) = E(H)/(V(H)  1) be the
density of H. G is said to be uniformly dense if and only if G has
the maximum density among all nontrivial subgraphs of G. The
concept of uniform density has been defined for matroids and studied
extensively by Catlin, Grossman, Hobbs and Lai. These graphs were
studied in a different setting by Bruno and Wienberg, Tomizowa,
Narayanan, Fujishige and some others. Computing the densities g of
subgraphs of a graph play a key role in analyzing the survivability
of a graph or a network (by Cunnigham, Gusfield, Hobbs), and in
examining electrical simulations (by Kishi and Kijitani).
Constructing uniformly dense graphs turn out to be useful in many
practical situations. In the talk, I will present a systematic
method of progressively modifying a given graph to obtain a
uniformly dense graph on the same number of vertices and edges. The
complexity of the algorithm will also be discussed. This is a joint
work with HongJian Lai and Hongyuan Lai.
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