Math 6161- Algebraic Combinatorics :
|Gambelli proved in 1902 that any
algebra that has a basis that satisfies a Pieri like multiplication rule is
isomorphic to a quotient of the symmetric functions. For this reason questions
in many seemingly unrelated fields can be reduced to statements in this algebra
and resolved using computational techniques that we use to study the symmetric
functions. The algebraba of symmetric functions is fundamental because
it provides a common language in which to understand questions in geometry,
combinatorics, algebra, and representation theory.
This course begins with an introduction to the representation theory of
the symmetric group to provide some motivation and introduce notation.
The symmetric functions are defined as a Hopf algebra and the structure of
this algebra is studied from a combinatorial perspective by examining the
fundamental bases and the change of basis coefficients. Only later is this
algebra shown to be isomorphic to the ring of polynomials which are invariant
under permutations of the variables. The use of Maple to do computation
will be encouraged by the introduction of the package SF by John Stembridge.
| This course meets twice a week, Tuesday/Thursday
from 2:30-5:30 in S101A Ross.
Part of this time will be lecture/problem session and we will spend part
of this time in a computer lab (where depends on the number of students attending
The representation theory component of the course will follow "The Symmetric
Group : Representations, Combinatorial Algorithms & Symmetric Functions"
by Bruce Sagan. The part of the course that covers symmetric functions
will follow notes that I am in the process of writing that approach the subject
of symmetric functions from the plethystic notation perspective.
|Prof. Mike Zabrocki
|Office: Ross S615 (NOTE this will change soon)
|Office Hours: by appointment
|Homework and Labs: 30%
|Final Exam: 40%
Check the schedule for dates of the midterm/homeworks/final.