# Thomas Pietraho

### Research Interests

- Representation theory and related combinatorics.

**About the image: **Machine learning models can sometimes reproduce results from pure mathematics; a challenge is interpreting the mechanism by which they do this. Above is a visualization of a hidden layer from a neural network that has learned how to multiply permutations in the symmetric group **S _{5}** . To compute the product

*c= ab*, the network learns the left coset of

*c*with respect to each of the six dihedral subgroups of order 10. Since these subgroups intersect trivially,

*c*is identified uniquely. Each image in the sequence represents a left coset partition of

**S**with respect to a different dihedral subgroup; each petal a different permutation.

_{5}### Courses

Please see the courses website for links to additional courses.

### Publications

Drafts of and links to the following articles are available on my personal web page.

- Orbital Varieties and Unipotent Representations of Classical Semisimple Lie Groups, PhD Thesis, MIT.
- Components of the Springer Fiber and Domino Tableaux,
*Journal of Algebra*, 272 (2):711-729, 2004. arXiv:math.RT/0210416 - A Relation for Domino Robinson-Schensted Algorithms,
*Annals of Combinatorics*, 13 (4):519-532, 2010. arXiv:math.CO/0603654. - Orbital Varieties and Unipotent Representations Attached to Spherical Orbits, arXiv:math.RT/0603685
- Cells in the Weyl Groups of type B(n),
*Journal of Algebraic Combinatorics*, 27(2):247-262, 2008. arXiv:math/0607231 - Cells and Constructible Representations in Type B,
*New York Journal of Mathematics*14:411-430, 2008. arXiv: 0710.3846 - Knuth Relations for the Hyperoctahedral Groups,
*Journal of Algebraic Combinatorics*, 29(4):509-535, 2009. - Module Structure of Cells in Unequal Parameter Hecke Algebras, Nagoya Mathematical Journal, 198(2010). arXiv:0902.1907
- Sign under the domino Robinson-Schensted maps, Annals of Combinatorics 18 (2014). arXiv:1301.1356.
- On the Sign Representations for the Complex Reflection Groups G(r,p,n) (with A. Mbirika and W. Silver),
*Beitrage zur Algebra und Geometrie*57, 2016. arXiv:1303.5021. - Kazhdan-Lusztig left cells in type B for intermediate parameters (with E. Howse). arXiv:1902.09301.

### Education

- PhD, Massachusetts Institute of Technology
- MS, University of Chicago
- BA, University of Chicago