Tableaux Reading Group, Spring 07
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



Reading
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
Read
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 411
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 1124
(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 25May 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 Grepresentation,
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 5May 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

910,1112, 13


910,1112, 13 

910,1112, 13, 34



yes





4

2

Jason

not

1112, 34

1112, 35

1112, 34

35

1112,34



yes 




else

3

Jeff

not








yes 




123, 4

4

Robert Gysel

not

1012

1112

1012

1011

1012



yes 




pm

5

Robert Gutierrez 
not 







yes 





6

Tom Denton 
not 







yes 





Dates MV is away
April 2007:
Apr 1213 jury duty!
Apr 14, conference at SFSU
Seattle, UW, Apr 1820.
AMS 2007 Spring Western Section Meeting Tucson, AZ, April 2122, 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
711, 2007
School: Algebraic Geometry and Algebraic Combinatorics May 2125, 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