MAT 246             University of California, Davis                 Winter 2011

Homework 3
due Feb 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.

°
(that list may well change as I fine tune the HW)

1.     7.20a (p 454)
2.     7.24ab (optional cde) (p 456)     Please WRITE this one up to hand in.
3.     7.27ce (p 457)     Please WRITE up 27c to hand in.
4.     7.28c (p 458)     AND find a (similar) product expansion for ∑λ s(2 λ)
Also, if you look at 7.28a, note that when A is a permutation matrix, this is related to the number of fixed points.

5.     Please WRITE this one up to hand in.
Find a determinental formula for pn in terms of ek's. In other words construct a matrix all of whose entries are either 0 or some ek (where k depends on i,j if it's the i,j entry), and so its determinant gives pn. [Hint: looking at their generating functions may help]

6.     Calculate the Schur function s(2,1) (x1, x2, x3) as the quotient of two 3×3 determinants a(4,2,0) / a(2,1,0) .

7.     We'll say a polynomial or function f is alternating (or anti-symmetric or skew-symmetric) if (i,j) . f = - f for all i ≠ j. (i.e. , equivalently f(... xi, xi+1, ... ) = -f(... xi+1, xi, ... ) .)
(a) If f and h are alternating but g is symmetric, show fg is alternating, and show fh as well as f/h (if f/h happens to be a polynomial or function again) are symmetric.
(b) Just as mλ is the symmetrization of the monomial xλ, show aλ is the anti-symmetrization of the monomial xλ (taken over a finite number of variables, usually the length n of λ), and that these form a basis of all alternating polynomials.
(c) Expand ∑σ ∈ Sn aδ + σ(λ) in that basis, where λ= (5,4,1) and n=3.

8.     Suppose σ = a1 a2... an ∈ Sn is a permutation in one-line notation such that P = P(σ ) has rectangular shape (under RSK). Let the complement of σ be σc = b1 b2... bn ∈ Sn where bi = n+1−ai for all i.
(a)
Also define the complement Pc of a rectangular standard tableau P with n entries to be the array obtained by replacing P(i,j) with n+1−P(i,j) for all (i, j) and then rotating the result by 180 degrees. Show that P(σc) = (Pc)T.

9.     For any symmetric polynomial f, let f be the operator adjoint to multiplication by f with respect to the Hall inner product, that is, < f g, h > = < g, fh > for all g,h ∈ Λ.
(a) Find a formula for hk mλ , expressed again in terms of monomial symmetric functions mμ
(b) Show that the basis of monomial symmetric functions is uniquely characterized by the formula from the previous part.