The Algebra and Combinatorics seminar is devoted to studying algebra,
combinatorics, their interconnection, and their relations to mathematics
and applications. Please visit our
Research
Page.
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Date
Time |
Location | Speaker |
Title – click for abstract |
 |
09/04
3:00pm |
MILN 317 |
|
Organizational Meeting |
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09/11
3:00pm |
MILN 317 |
Michaela Vancliff
UT-Arlington |
Generalizing Graded Clifford Algebras and their Associated Geometry
Abstract: Graded Clifford algebras are non-commutative algebras that have been known since at least the 1980s, and one can read off certain properties of such an algebra from certain commutative geometric data associated to the algebra. In particular, a standard result is that a graded Clifford algebra C is quadratic and regular if and only if a certain quadric system associated to C is base-point free. Recently, T. Cassidy and M. Vancliff introduced a generalization of such algebras, so called graded skew Clifford algebras, and they found that most of the results concerning graded Clifford algebras can be extended to graded skew Clifford algebras, providing the appropriate non-commutative geometric data is defined. This interaction between algebra and geometry has led to the construction of new quadratic regular algebras of global dimension four, thereby contributing to the work of classifying all such algebras. In this talk, the above algebra-geometry correspondence will be our main focus, together with the history concerning it, and the setting of graded skew Clifford algebras will be introduced along with many examples. |
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09/18
3:00pm |
MILN 317 |
Mahir Can
Tulane |
Gelfand pairs and parking functions
Abstract: Let H ⊆ G be a subgroup. The pair (G,H) is called a Gelfand pair, if the convolution algebra of H-invariant functions on the symmetric space G/H is commutative. Equivalently, the induced trivial representation 1GH is multiplicity-free. In this talk, we: - first analyze (for the heck of it) the well known Gelfand pair: (Bn,Sn) the hyperoctahedral group and its symmetric subgroup Sn, then we evaluate its spherical functions.
- study a Gelfand pair related to the parking functions. We show that the number of irreducible constituents of the induced representation is the Catalan numbers.
This is joint work with Kursat Aker (Feza Gursey Institute, Turkey). |
|
09/25
3:00pm |
MILN 317 |
Kate Owens
South Carolina |
Finite Axiomatizability and Commutative Directoids
Abstract: An algebra is a nonempty set equipped with some finitary operations. The equational theory of an algebra is the set of all equations true in that algebra. If we can deduce all of an algebra's true equations from a finite set of equations true in the algebra, we say that the algebra's equational theory is finitely axiomatizable. Ježek and Quackenbush devised a way to convert any up-directed partially ordered set into an algebra by imposing a two-place operation on the set which always outputs a common upper bound of its inputs and, in the case of comparable inputs, will output the larger. The resulting algebras are called directoids. In this talk we examine recent work on the finite axiomatizability of the equational theory of commutative directoids.
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10/02
2:30pm |
MILN 317 |
Sarah Mason
UCSD, Wake Forest |
Multiplication rules for modules over the ring of symmetric functions
Quasisymmetric Schur functions are a new basis for quasisymmetric functions. In joint work with Haglund, Luoto, and van Willigenburg, we provide a combinatorial interpretation for the positive structure constants obtained when these functions are multiplied by Schur functions. This result plays a role in the proof (joint with Lauve) that Bergeron and Reutenauer's conjectured basis is indeed a basis for the quotient ring of quasisymmetric functions over symmetric functions. |
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10/05
3:00pm |
MILN 216 |
Josephine Yu
MSRI |
Linear systems on tropical curves
(—Joint with —)
A tropical curve is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system |D| of a divisor D on a tropical curve analogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, |D| defines a map from the tropical curve to a tropical projective space, and the image can be extended to a parameterized tropical curve of degree equal to deg(D). The tropical convex hull of the image realizes the linear system |D| as an embedded polyhedral complex. We also show that curves for which the canonical divisor is not very ample are hyperelliptic. This is joint work with Christian Haase and Gregg Musiker. |
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10/09
3:00pm |
MILN 317 |
Aaron Lauve
Texas A&M University |
Skew Littlewood–Richardson rules from Hopf algebras
We use the rational action of a Hopf algebra on its dual to
study products of skew Schur functions in the ring of symmetric
functions. The result is a version of the Littlewood-Richardson rule for
skew Schur functions that simplifies, and proves, a conjecture of
Assaf and McNamara (recent preprint). We also establish similar skew
Littlewood-Richardson formulas for Schur P- and Q-functions, and other families of functions in algebraic combinatorics. |
|
10/23
3:00pm |
MILN 317 |
Eva Y.P. Deng
Dalian University of Technology, China |
Some Identities on the Catalan, Motzkin and Schröder Numbers
Catalan, Motzkin and Schröder Numbers are closed related. In
this talk, we present several identities involving the Catalan, Motzkin and
Schröder numbers by using Riordan group. Furthermore, we introduce the
symmetric Dyck paths, symmetric Motzkin paths and symmetric Schröder
paths; the relations between their cardinalities are shown too. Finally, we
give some combinatorial proofs of above identities. This is joint work with
L.H. Deng, L.W. Shapiro and W.J. Yan. |
|
10/30
3:00pm |
MILN 317 |
Katia Consani
Johns Hopkins |
Schemes over F1 and zeta functions
The talk will be an overview on the recent developments of the
theory of F1-schemes and the arithmetic applications on the description of the counting function of the expected ‘curve’ over F1, whose zeta function is the (complete) Riemann zeta function. |
|
11/06
3:00pm |
MILN 317 |
Jon Kujawa
Oklahoma |
Varieties and Lie Superalgebras
As we learned from Descartes, even if you are only
interested in algebra questions, you can gain an awful lot of
insight by looking at the related geometry. Following a long
history in representations of finite groups and similar
settings, I along with Boe and Nakano introduced varieties
whose geometry tells you interesting things about the
representation theory and combinatorics of Lie Superalgebras
over the complex numbers. We will introduce these varieties
and discuss some of our results and conjectures. The talk
will start from scratch and should be accessible to grad
students and non-specialists. |
|
11/09
3:00pm |
MILN 216 |
Gregg Musiker
MIT |
TBA, joint with Algebraic Geometry Seminar |
|
11/13
3:00pm |
MILN 317 |
Ke Ye
Texas A&M University |
The stabilizer of immanants
Immanants are polynomials of degree n in n2 variables associated irreducible representations of the symmetric group on n elements
Sn. We describe immanants as trivial Sn
modules and showed that any homogeneous polynomial of degree n on the
space of n×n matrices preserved up to scalar by left and right
action by diagonal matrices and conjugation by permutation matrices is a
linear combination of immanants. M. Antónia Duffner found equations
that determine the stabilizer of immanants (except determinant and
permanent) in the group GL(E⊗F). We solve these equations and
give the explicit description of the stabilizer. |
|
11/20
3:00pm |
MILN 317 |
Simon Guest
Baylor |
A Solvable Version of the Baer–Suzuki Theorem
Let G be a finite group, and take an element x in G. The Baer–Suzuki
states that if every pair of conjugates of x generates a nilpotent group
then the group generated by all of the conjugates of x is
nilpotent. It is natural to ask if an analogous theorem is true for solvable
groups. Namely, if every pair of conjugates of x generates a solvable group
then is the group generate by all of the conjugates of x solvable?
In fact, this is not true. For example, if x has order 2 in a (nonabelian)
simple group G then every pair of conjugates of x generates a dihedral
group (which is solvable), but the normal subgroup generated by all
of the conjugates of x must be the whole of the nonabelian simple group
G, which of course is not solvable. There are also counterexamples when x
has order 3. However, the following is true:
1. Let x ∈ G have prime
order p ≥ 5. If every pair of conjugates of x generates a solvable group
then the group generated by all of the conjugates of x is solvable.
2. Let x ∈ G be an element of any order. If every 4-tuple of conjugates x,
xg1, xg2,
xg3 generates a solvable group then the group
generated by all of the conjugates of x is solvable.
We will discuss these results, some generalizations, and some of the methods used in their proof. |
|
12/04
3:00pm |
MILN 317 |
Lars Kadison
UPenn & UC San Diego |
Depth of a subgroup or when group subring is "normal" in tower of iterated endomorphism rings
The depth > 1 of a subalgebra pair of semisimple algebras B < A may
be viewed as the smallest value n where B < An-2 is normal subring
(as defined by Rieffel), equivalently depth two subring, where A1 is End AB,
and A2 is End of A1 as right A-module, etc. (A depth one ring extension is
centrally projective. By a result of Boltje-Kuelshammer to be discussed, a depth
two Hopf subalgebra is normal. ) The depth may be read off the inclusion diagram
of B in A, or by comparing the number of zero entries in the powers of the
inclusion matrix. Finally, via Mackey theory a subgroup H in a finite group G
has depth 2n or less if its core is the intersection of n conjugates of H; if
corefree, depth less than or equal to 2n-1. |
