Student Research

Macalester College > Academic > Math/CS > Tom Halverson > Research

Tom Halverson
Collaborative Research Projects with Macalester College Students

The following research has been done in collaboration with Macalester College undergraduate students. This research was supported, in part, by Macalester College and by National Science Foundation Research at Undergraduate Institutions Grants (NSF-RUI) DMS-9800851, DMS-0100975, and DMS-0401098. Some of this work has led to publications that are listed on my research page. All of these projects have led to honors and capstone papers at Macalester College.

If you are on this list, please send me an update of what you are doing.

2007 - 2008

Anne Moore, Macalester College class of 2008.

Representations of the Temperley-Lieb Algebra

Using the fact that the Temperley-Lieb algebra is a quotient of the affine Hecke algebra, we find the seminormal representations of the Temperley-Lieb algebra, on a basis indexed by 2-row standard tableaux. Our method is to push the action of the Hecke algebra through the surjection and then verify the Temperley-Lieb relations.

2006 - 2007

Owen Anderson, Macalester College class of 2008.

Random Walks on Set Partitions

We study random walks on set partitions given by randomly transposing elements with transpositions in the symmetric group. We also study random walks on these set partitions by randomly choosing generators of the partition algebra and multiplying. In each case we observe the cutoff phenomenon and we analyze the steady state and the rate of convergence.

2005 - 2006

Jeffrey Barnes, Macalester College class of 2007.
Tensor Power Centralizer Algebras for Rank 2 Complex Reflection Groups

For each finite subgroup G of the special unitary group SU(2), we show how to construct a tower of finite-dimensional tensor power centralizer algebras. We especially study this construction for the binary tetrahedral, octahedral, and icosahedral groups T, O, and I. The Bratteli diagram for the case of the icosahedral group can be seen here.

Michael Decker, Macalester College class of 2006.
Explicit Combinatorial Models for Multiplicity Free Representations of Diagram Algebras

We study a family of semisimple diagram algebras: the partition algebra, the half-integer partition algebra, the Brauer algebra, the Temperley-Lieb algebra, the planar partition algebra, the rook monoid, the planar rook monoid, and the symmetric group. We show how to uniformly and combinatorially define an action on certain diagrams that leads to a representation of each of these algebras. We conjecture that each irreducible representation of these diagram algebras appears in this model with multiplicity 1.

Kate Herbig, Macalester College class of 2006.
Combinatorial Repesentation Theory of the Planar Rook Monoid

The planar rook monoid is a the sub monoid of the rook monoid of diagrams that can be drawn without edge crossing. It is also the set of one-to-one functions whose domain and codomain are subsets of {1, 2, ..., k} and which satisfy f(i) < f(j) if i < j. We find a presentation on generators and relations. We determine a Schur-Weyl dualtity with GL(1,C). We explicitly compute irreducible relations, show that the Bratteli diagram is Pascal's triangle, and compute the character table.

2004 - 2005

Michael Decker, Macalester College class of 2006.
A Bitrace for the Finite Symplectic Group and the Iwahori-Hecke Algebra of Type C.

We study the simultaneous trace of the finite symplectic group G=Sp(2n,q) and its Iwahori-Hecke algebra HC(n,q) acting on the flag variety C[G/B], where B is the Borel subgroup of upper trianglar matrices. We have computed this bitrace for many special cases and have studied the recursive structure for the general formula.

2003 - 2004

Ian McCowan, Macalester College class of 2004.
Up-Down Tableaux and Characters of the Partition Algebra.

This research analyzes the combinatorics of up-down or vacillating tableaux with an aim to finding recursive Roichman-like weights on vacillating tableaux that can be used to compute the irreducible characters of the partition algebra. In the special case of the symmetric group (which lives inside the partition algebra), we show that the classical Murnaghan-Nakayama rule can be derived from the Roichman rule, but that in fact the Murnaghan-Nakayama rule is computationally more efficient. In the case of the partition algebra, the Murnaghan-Nakayama rule is known but the Roichman rule is not. We show what the Roichman weights must be, in many special cases, and give other partial results. Ian presented his research at the Undergraduate Research Poster Session at the joint AMS-MAA Meetings in Phoenix in January 2004.

2002 - 2003

Hannah Ferber, Macalester College class of 2003.
Combinatorics of the Party Algebra

Kathryn McNally, Macalester College class of 2003.
Combinatorics of the Party Algebra

Tim Lewandowski, Macalester College class of 2003.
RSK Insertion for the Partition Algebra

Ben Sherwood, Macalester College class of 2003.
Enumeration of GL(n,q)/B Cosets

2001 - 2002

Andy Cantrell, Macalester College class of 2002.
RSK Insertion and Characters of Cyclotomic Hecke Algebras

In collaboration with Brian Miller (below), Andy determined an RSK insertion algorithm that combinatorially describes Shoji's Frobenius formula for the cyclotomic Hecke algebras. He then used the formula to determine Roichman weights for the irreducible characters of these algebras. These formulas specialize to the cyclotomic groups, which are wreath products of symmetric groups with finite cyclic groups. Andy is now a graduate student in astronomy at Yale (link) and a practicing potter. Andy presented his research at the Undergraduate Research Poster Session at the joint AMS-MAA Meetings in San Diego in January 2002. He also gave a talk at the same meeting, in the Undergraduate Research Special Session.

Brian Miller, Macalester College class of 2002.
Standard Tableaux and RSK Insertion for the Partition Algebra

Brian's research contributed both to the RSK insertion for cyclotomic Hecke algebras, which led to the paper with Andy Cantrell and Tom Halverson (see the research page), and to the RSK insertion for the partition algebra (see the Halverson-Lewandowski paper also on the research page). Brian is now a successful Irish musician and member of 5 Mile Chase.

Bill Owens, Macalester College class of 2002.
Representations of the Partition Algebra

Bill Owens computationally analyzed the representation of the partition algebra on tensor space. Using these tools, he (amazingly) conjectured the correct form of the Murphy element for the partition algebra and proved that the sum of the Murphy elements is in the center. Here is one of Bill's Murphy elements:
murphy1

2000 - 2001

Chris Bremer, Macalester College class of 2001.
The Chevalley-Hecke Bitrace for the M/B, where M is the full Monoid of Matrices over a Finite Field.

John Farina, Macalester College class of 2001.
An Orthogonality Relation for the Parition Algebra

1999 - 2000

Momar Dieng, Macalester College class of 2000.
Characters of the q-Rook Monoid Algebra

1998 - 1999

Erik Fuller, Macalester College class of 1999.
Enumerating Irreducible Representations of Affine Hecke Algebras

Alex Mas, Macalester College class of 1999.
Combinatorics of the Chevalley-Hecke Bitrace

Vahe Poladian, Macalester College class of 1999.
Characters of the Rook Monoid

1996 - 1997

David Castro, Macalester College class of 1997.
Seminormal Representations and Murphy Elements of the Walled Brauer Algebra

Aaron Mohrman, Macalester College class of 1997.
Second Orthogonality Relations for the Walled Brauer Algebra

1995 - 1996

Michael Wolfe, Macalester College class of 1996.
Chromatic Symmetric Functions

This research has appeared in Pi Mu Epsilon Journal, 10, Spring 1998, 643-657.

An element of B6