MAT 246             University of California, Davis                 Winter 2011

Homework 2
due Jan 28?, 2011 , in class

Haven't quite figured when is the most sensible due date for this. Let's say Jan 28 for now, subject to postponement (or adding of exercises). In fact, I think it's quite likely I'll add some exercises. Maybe some that I'll make up myself.

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!

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 7.9, 7.10, 7.11, 7.12 , 7.14, 7.15
(that list may well shorten as I fine tune the HW)

0.     7.12     which was postponed from last set. (some likelihood i'll ask this to be written up)
1.     7.13a (p 451)
2.     7.3 (p 450)     (i think this was a typo to repeat it in hw2)
3.     7.15 (p 452)     probably i'll ask you to write up at least part (a), which i like, but ONLY if i can figure it out 1st and give some good hints
4.     7.16a (p 452)     (Read 16b,e for fun with Catalan numbers)
5.     7.17 (p 453)
6.     7.20a (p 454)    
7.     (WRITE up, please)     Let ∂k be the operator on symmetric functions given by partial differentiation with respect to pk under the identification of Λ with Q[p1, p2 ...]. Show that ∂k is adjoint to the operator of multiplication by pk/ k with respect to the scalar product.

8.     (WRITE up, please)     Use RSK to show
    (a) A permutation σ is an involution iff P(σ) = Q(σ) , where (P,Q) is the image of σ under RSK.
    (b) The number of involutions in Sn is ∑ |λ| = n fλ
    (c) Write a product formula for ∑λ sλ

9.     (NEW problem, added Jan 19)
(a)     Let a = ∏ i ≠ j (1 - xi/xj). When working over n variables, what is the constant term of a? Call that a(n).
(b)     For n=2, compute the constant term of m11 h11 a.
(c)     For n=2, compute the constant term of m2 h11 a.
(d)     For n=2, compute the constant term of m11 h2 a.
(e)     For n=2, compute the constant term of m2 h2 a.
(f)     Can you argue in general (general n) that the constant term of mλ hμ a = a(n) δλ μ ?