MAT 246             University of California, Davis                 Winter 2011

Homework 4
due Feb 25, 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 7.15, 7.16, 7.17, and A1 Appendix (maybe 3.4 and 3.5 of Sagan too)
(that list may well change as I fine tune the HW)

1.     7.30ab (p 459)
2.     7.35ab (optional c) (p 460)     Please WRITE this one up to hand in.
3.     7.36 (p 460)
4.     7.41 (p 461)     (you might want to read 7.42 for interest)

5.     Please WRITE this one up to hand in.
Let A be an n × n matrix. Let cA(t) be the characteristic polynomial of A. Let ak = tr(Ak). Write the coefficients of cA(t) in terms of the ak. I'll start you out-- the leading coeff is 1 which we can think of as 1/n*a0; the coeff of tn-1 is +/- a1. And so on...

6.     Show Δ is co-associative. That is, as maps (1 ⊗ Δ) ° Δ = (Δ ⊗ 1) ° Δ where Δ : Λ -- > Λ ⊗ Λ = Λ(x) ⊗ Λ(y) is given by Δ(f) = f(x,y).     (1 is the identity map above.) (Think about how co-associativity looks/feels like the associativity of multiplication/usual product if you were to write that out formally.)

7.     Recall Young's lattice of partitions where we'll write μ ⊂ λ if μ is contained in λ and the skew shape λ/μ contains 1 box. Define operators D and U by D(λ) = ∑μ ⊂ λ μ and U(λ) = ∑λ ⊂ ν ν . Show
(a)     DU - UD = I     (look familiar?!!) and
(b)     DUk = k Uk-1 + UkD.
(c)     Derive anew that ∑|λ|=n ( fλ)2 = n!