Abstract Nonsense

Crushing one theorem at a time

Current Schedule:Summer REU and Blogging


I thought that I would give any (if there exists any) regular readers of my blog a heads-up as to what is coming up in the following months in terms of blogging.

This summer I will be attending the SMALL program at Williams College in Williamstown, MA. Specifically I shall be working in the Algebraic and Geometric Combinatorics project with Elizabeth Beazley.

In Dr. Beazley’s own words the project is as follows:

“The affine symmetric group is an infinite analog of the group of permutations on n letters.  The quotient of the affine symmetric group by the finite symmetric group is called the affine Grassmannian, which arises in a myriad of mathematical contexts, including algebraic geometry, representation theory, number theory, and both algebraic and enumerative combinatorics.  There are many different combinatorial models for elements in the affine Grassmannian:  in terms of core diagrams, bounded partitions, abacus diagrams, points in a root lattice, and alcoves in a hyperplane arrangement.

The affine symmetric group is one example in the more general family of infinite Coxeter groups, and one can define more general affine Grassmannians by taking analogous quotients of other infinite Coxeter groups.  There are also appropriate adaptations of the notions of cores, bounded partitions, abaci, root lattice points, and alcoves.  This summer we will explore a variety of combinatorial and geometric questions concerning the various combinatorial models for these other types of affine Grassmannians.”

Consequently, I will be spending the next (roughly) ten weeks blogging about prerequisite things to understanding/researching the topics mentioned above. In particular, I will be talking a lot about reflections groups, Coxeter groups, representation theory, and some other geometric/combinatoric things which will mainly come from the following books/papers (my work-in-progress bibliography):

Books:

  1. Humphreys, James E. Reflection Groups and Coxeter Groups. Cambridge: Cambridge UP, 1990. Print.
  2. Björner, Anders, and Francesco Brenti. Combinatorics of Coxeter Groups. New York, NY: Springer, 2005. Print.
  3. Aigner, Martin. A Course in Enumeration. Berlin: Springer, 2007. Print.
  4. Sagan, Bruce Eli. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. New York: Springer, 2001. Print.
  5. Manivel, Laurent. Symmetric Functions, Schubert Polynomials, and Degeneracy Loci. Providence, RI: American Mathematical Society, 2001. Print.
  6. Humphreys, James E. Introduction to Lie Algebras and Representation Theory. New York: Springer-Verlag, 1972. Print.
  7. Fulton, William, and Joe Harris. Representation Theory: A First Course. New York: Springer-Verlag, 1991. Print.

Papers:

  1. Chris Berg, Brant Jones, and Monica Vazirani.  A Bijection on Core Partitions and a Parabolic Quotient of the Affine Symmetric Group, J. Combin. Theory Ser. A  116 (2009), no. 8, 1344-1360.
  2.  Christopher Hanusa and Brant Jones, Abacus Models for Parabolic Quotients of Affine Weyl Groups, math.CO/1105.5333v1, 2011.
  3. Luc Lapointe and Jennifer Morse, Tableaux on k + 1-cores, Reduced Words for Affine Permutations, and k-Schur Expansions, J. Combin. Theory Ser. A 112 (2005), no. 1, 44-81.

Perhaps when I get bored/have extra time I shall get back on track with the blogging I was currently in the middle of (but got sidetracked) meant to prime me for the classes I am taking next term: Algebraic Number Theory, and Complex Analytic Geometry. In particular, I will be blogging about complex analysis and Galois theory. But, like I said, this is not going to happen until the end of summer.

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June 10, 2012 - Posted by | Uncategorized

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