MAT 246             University of California, Davis                 Winter 2011

Homework 5
due Mar 11, 2011 , in class

I'm going to list several to start on and think about. I'll pick out some subset to write up nicely and hand in. Without a reader, I'm not sure how many problems will get graded. But you should do many hw problems to learn the material, regardless of whether they are collected!
And if you've taken the time to write up more than what's required to hand in, feel free to turn that in too--especially if I may be lazy and use that as the posted solution (I'll certainly factor those extra efforts into your grade).

    Make sure your homework paper is legible, stapled, and has your name clearly showing on each page.    Also, for alphabetizing purposes, in the upper LeftHand corner of your paper please put the first letter of your last name, large and circled.  

No late homeworks will be accepted! (but good arguments to shift the due date might be)

Collaboration with classmates is encouraged, so long as you write up your solutions in your own words. Please note on the front page of your HW who you have collaborated with.    

Read EC II, Sections 7.17, 7.18 (and you should have already read A1 Appendix and maybe 3.4 and 3.5 of Sagan too) other sections of Sagan can be helpful too

This set lists several problems so looks LONG, but that's because we are about to end the quarter. Many of them are optional, or it's just suggested you read the statement of the questions to know it's true and out there. So, this set has only 10 real problems, but they are numbered 1-15 for organizational purposes.

1.     7.34 (pg 460)     OPTIONAL (this one may help in solving 7.35c)
2.     7.37a (p 460)    
3.     READ 7.40 (pg 461)    
4.     READ 7.47a (pg 462-3) for an interesting way to connect graph theory and symmetric functions    

5.     7.50 (pg 466)     Please WRITE this one up to hand in.
6.     7.52 (pg 466)     Please WRITE this one up to hand in.

7.     7.59 (pg 467) OPTIONAL , to learn more about border strips and hooks and p-cores (and maybe p-quotients)
and if you've done 7.59, then it motivates a bit why 7.60 (pg 469) might be an interesting result or why it is true from another pov

8.     7.67a (pg 471)     (optional to think about bcd)     Also, think about the connection btwn this problem and the fact that the pn are the primitive elements of Λ as a Hopf algebra.     Please WRITE part (a) up to hand in.

9.     7.68a (pg 471-2) OPTIONAL, for those who know representation theory of finite gropus
10.     7.69a (pg 472)
11.     7.71c (pg 474)     Parts a,b are optional, for those who are comfortable with representation theory of groups; you may use parts a,b to do c, even if you did not complete a,b

12.     Using the Murnaghan-Nakayama rule, make/calculate the table χλ(μ) for all partitions of 4.

13.     Let V be the n-dimensional representation of Sn defined by σ( ei ) = eσ(i) ; where { ei } is the standard basis of V. Which symmetric function does its character correspond to under the map ch?     Please WRITE this one up to hand in.

14.     Let |λ|=n.     Show that < hλ, h(1n) > is the number of cosets of the Young subgroup Sλ in the symmetric group Sn.     Better yet, show it using representation theory--what is the above computing on the level of characters?
    OPTIONAL: What similar thing is < hλ, hμ > counting?

15.     Similar to what we did in HW3, # 5 , you can write er = a determinant involving pk's. On the other hand, the Murnaghan-Nakayama rule writes s(1r) in terms of pλ's. Show the two formulas agree.