who:
 Monica Vazirani vazirani 3224 Matthew  Register jmregist 1 Jason Hole jhole 2 Jeffrey Paul Ferreira jferreira 3 Robert Gutierrez matico1982 4 Robert Simon Gysel gysel 5 Tom Denton sdenton 6 (Anne Schilling) anne (Steve Pon) spon (Michelle Stutey) mstutey

The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions,
Second Edition (Graduate Texts in Mathematics)   by Bruce E. Sagan (Author)

and

Young Tableaux (London Mathematical Society Student Texts)  by William Fulton

after a bit, we'll get to crystals (the crystal structure on SSYT and another way of seeing knuth equivalence) and/or some research papers.

Assignments

week of Apr 2
let's try all starting w/ page 79 of fulton, then try jumping in to ch 7.
next ch 8, then we can go back and fill in ch 6, ch 4, section 2.2, ch 5.

and/or that means in sagan, following along w/ ch 2.  (sagan has a bit more in his ch2 than fulton's ch 7).

since fulton is a bit sparser, my idea was to start there, and if we want to go into more depth, we can turn to sagan.

(if you need a brush up on reprentation theory, see sagan ch1, especially 1.3, 1.8, 1.12)

Exercises

week of Apr 2
do whatever exercises you run across in fulton.
pg 86,87, sagan  try exercises 1, 4, 7, 10, 12, but i think they're harder than the fulton ones, so stick w/ fulton to start.

week of Apr 4-11
read fulton 7.2 and 7.3, doing exercises along the way. Also, try the porridge/knights question and the martian/cube question, more formally.

weeks of Apr 11-24
(fulton 7.4 is optional, but there are some good exercises, like the one over F_3 if you like algebra.) read fulton 8.1 all about the GL_n side, tensors, exterior powers, symmetric powers, doing exercises along the way. (fulton+harris may have a nice exposition in the first chapter or the appendices.)
Meet Friday Apr 27 at 4pm.

week of Apr 25-May 4
continue reading fulton 8.1 all about the GL_n side, tensors, exterior powers, symmetric powers, doing exercises along the way. (fulton+harris may have a nice exposition in the first chapter or the appendices.) Continue with 8.2 (even 8.3) supplementing in Sagan. Rob (and others interested) are going to the start of Sagan to get down the basics.
Tom has provided a link to a good exposition http://darkwing.uoregon.edu/~brundan/math647fall04/index.html
I have also made up a problem sheet to guide you through the porridge problem more systematically. pdf   possibly this copy is clearer? pdf1 . Another way to think about what is going on -- as a representation of G, the state space is multiplicity free (G= C n or S 4 ), and the knight's/martians' actions commute with G. By Schur's Lemma, the associated matrix acts as a constant on each irreducible component. The magnitude of that constant controls the long term behavior as the action is repeated. (This POV also explains why commuting matrices (that are diagonalizable) share a common eigenbasis.)
Another fruitful POV to take is that having the knights' eigenspaces in hand gives you a way of decomposing the space as a G-representation, easily. (Try is for S 4 ).
Take a look at Sagan's exercise 12 for some fun properties of S n 's character table.
Exercise for 8.1: let V = C 4. Decompose the 3rd tensor prod of V (V.V.V) as a GL 4 representation. Use the
E lambda's of 8.1 . You should be able to write it out (as subrepresentations of V.V.V) explicitly for lambda=(3) and lambda =(1,1,1). What about lambda = (2,1) ? What is the dimension of that subspace? [What if dimV=2? if dim V = 3? if dim V = 5?] What is the trace of a general diagonal matrix w/ entries x 1 ... x 4 on each subspace?
8.3 explains why all the GL n stuff looks a lot like S d combinatorics as in Sagan.
Meet Friday May 4 at 4pm.

week of May 5-May 14

We can try meeting Monday May 14 if people are free at 4pm (since I'm gone May 11).

Scheduling
4
 M T W R F MV 1 Matthew not 9-10,11-12, 1-3 9-10,11-12, 1-3 9-10,11-12, 1-3, 3-4 yes 4 2 Jason not 11-12, 3-4 11-12, 3-5 11-12, 3-4 3-5 11-12,3-4 yes else 3 Jeff not yes 12-3, 4-- 4 Robert Gysel not 10-12 11-12 10-12 10-11 10-12 yes pm 5 Robert Gutierrez not yes 6 Tom Denton not yes

Dates MV is away
April 2007:
Apr 12-13 jury duty!
Apr 14, conference at SFSU
Seattle, UW, Apr 18-20.
AMS 2007 Spring Western Section Meeting Tucson, AZ, April 21-22, 2007 (Saturday +- Sunday) Meeting #1027; Special Session on Algebraic Combinatorics

May 2007:
Centre de recherches mathématiques, Montreal.
Workshop: Combinatorial Hopf Algebras and Macdonald Polynomials May 7-11, 2007
School: Algebraic Geometry and Algebraic Combinatorics May 21-25, 2007, followed by Workshop: Interactions between Algebraic Combinatorics and Algebraic Geometry May 28 - June 1, 2007

June 2007:
June 4 to June 8, 2007, Arithmetic harmonic analysis on character and quiver varieties at the American Institute of Mathematics, Palo Alto, California