Math 665 Fall 2010
Symmetric Functions

Lectures: TTh 1:00-2:30

Instructor: Thomas Lam, 2834 East Hall,

Office Hours: Tuesday 11:45-12:30, Thursday 3-4

[EC2] Enumerative Combinatorics Volume 2, R. Stanley
[Mac] Hall polynomials and Symmetric functions, I. Macdonald
[Ful] Young tableaux, W. Fulton
[Sag] The symmetric group, B. Sagan

First Day Handout.

Problem sets: I will maintain a list of problems here, which will be updated as we cover more material. You can begin working on the problems at any time. Each problem set will be graded based upon your best 5 solutions. You can "save up" solutions for later, but you should start early!
The first problem set is due on September 30.
The second problem set is due on November 2.
The third problem set is due on December 7. Instead of submitting 5 problems, you may opt to submit 3 problems and a 3-5 page paper explaining a topic in symmetric functions. If you prefer the second option you must tell me before the due date of Problem Set 2 (and we can negotiate a topic).

For grading purposes, to be explained in class, I have indicated how many students have completed each problem.
\lambda' is the conjugate (or transpose) of a partition \lambda

  1. [9]Prove that \lambda >= \mu in dominance order if and only if \mu' >= \lambda' in dominance order.
  2. [8] Prove that dominance order corresponds to containment of convex hulls of S_m orbits. For example, (3,0,0) dominates (2,1,0) because the convex hull of (3,0,0),(0,3,0), and (0,0,3) contains (2,1,0),(1,2,0),...
  3. [9] Expand \prod_{i>=1}(1+x_i+x_i^2) in terms of elementary symmetric functions.
  4. Unsolved Let f: \Lambda \otimes R --> R be an R-algebra morphism such that f(s_\lambda)>=0 for all \lambda. Prove that the generating function a(t) = 1+f(h_1)t+f(h_2)t^2 + ... is meromorphic. (Bonus: all the zeroes of a(t) are real and negative.)
  5. [10] Let P(t) = p_1+ p_2 t + p_3t^2 + .... Prove that P(t)H(t)=d/dt H(t), and hence obtain a recursion which determines p_i's and h_i's in terms of each other.
  6. [6] Prove that \prod_i 1/(1-qx_i)\prod_{i,j} 1/(1-x_i x_j) = \sum_\lambda q^{c(\lambda)} s_\lambda, where c(\lambda) is the number of parts of \lambda' that are odd.
  7. [5] The sign sgn(T) of a SYT T is the sign of the reading word of T, obtained by reading the rows of T from left to right, starting with the top row. Show that sgn(w) = (-1)^{v(\lambda)} sgn(P) sgn(Q), where w <-> (P,Q) under RSK, and v(\lambda) is the maximum number of disjoint vertical dominoes that fit inside \lambda.
  8. Unsolved (It may be useful to first solve the previous problem.) Prove that \sum_{SYT T with size n} sgn(T) = 2^{[n/2]}.
  9. [3] A ribbon R is an edge-connected skew shape (the boxes in the Young diagram of R are connected by shared edges) which does not contain any 2x2 squares of boxes. Let \lambda be a partition. The n-core of \lambda is the partition \mu obtained from \lambda by successively removing ribbons of size n from \lambda while keeping the shape a Young diagram, until this is no longer possible. Show that the n-core of \lambda is well defined. Find the generating function for the numbers a_k of n-cores with size k.
  10. [5] Let \lambda be a partition of n with length r. The forgotten symmetric functions are defined as f_\lambda = (-1)^{n-r}\omega(m_\lambda). Show that the coefficient of m_\mu in f_\lambda is equal to the number of distinct permutations (\alpha_1,\alpha_2,...,\alpha_r) of (\lambda_1,...,\lambda_r) such that the set {\alpha_1+...+\alpha_i: 1\leq i \leq r} contains the set {\mu_1+...+\mu_j: 1\leq j \leq \ell(\mu)}.
  11. Unsolved A balanced tableau T is a filling of a Young diagram with the integers 1,2,...,n such that in each box b, the number T(b) is the k-th largest number in the hook of b, where k is the number of boxes below b, counting b itself. Prove that the number of balanced tableaux of shape \lambda is equal to f^\lambda.
  12. [8] Let \delta_n=(n-1,n-2,...,1). Prove that s_{\delta_n}(x_1,...,x_n)=\prod_{1<= i< j <= n} (x_i + x_j).
  13. [7] For w in S_n of cycle type \lambda, prove by a direct combinatorial argument that the number of permutations in S_n which commute with w is equal to z_\lambda.
  14. [6] Let x_n = 1/n^2 for n = 1,2,... It is known that E(-t^2) = sin(\pi t)/\pi t. Use this to find the value \zeta(2r) of the Riemann zeta function at even positive integers.
  15. [2] Show that (\sum_{n >=0} h_{2n+1})/(\sum_{n>=0}h_{2n}) is a power series in the odd power sums.
  16. [2] Prove that \sum_{\lambda even} s_\lambda = \prod_i 1/(1-x_i^2)\prod_{i < j } 1/(1-x_ix_j), where the sum is over all partitions \lambda with all parts even.
  17. Prove that \sum_{\lambda' even} s_\lambda = \prod_{i< j } 1/(1-x_ix_j). (It was pointed out by Seungjin that this question is trivially implied by Problem 6, so you can't get credit for both problems. Thanks, Seungjin.)
  18. [5] Prove that for a staircase shape \delta_n=(n-1,n-2,...,1), the Schur function s_\delta is a polynomial in the odd power sums.
  19. [4] Show that for f \in \Lambda, < f(x_1^r,x_2^r,...),s_\lambda >=0 unless the r-core of \lambda is empty.
  20. [7] Show that e_n = 1/n! det(u_{ij}), where
    u_{ij} = p_{i-j+1} if i>=j, u_{ij} = i if i+1 = j, and u_{ij} = 0 otherwise.
  21. [8] Let J_{mn} be the m x n matrix filled completely with 1's. Identify the image of J_{mn} under the RSK correspondence.
  22. [2] Find all symmetric functions f \in \Lambda, which have positive coefficients when expanded in terms of the {e_\lambda} basis, and also positive coefficients when expanded in terms of the {h_\lambda} basis.
  23. [1] Prove the following skew Pieri rule: h_k s_{\lambda/\mu} = \sum_r (-1)^r \sum s_{\rho/\nu} where the second sum is over all skew shapes \rho/\nu such that \rho/\lambda is a horizontal strip with (k-r) boxes, and \mu/\nu is a vertical strip with r boxes.
  24. [2] Let \lambda be a Young diagram with r boxes on the main diagonal. Prove that s_\lambda = det(s_{(\lambda_i-i+1,1^{\lambda'_j-j})})_{i,j=1}^r
  25. [1] Fix two partitions \alpha and \beta. Prove that \sum_\lambda s_{\lambda/\alpha}(x) s_{\lambda/\beta}(y) = \sum_\mu s_{\alpha/\mu}(x) s_{\beta/\mu}(y) \Omega(x,y) where \Omega(x,y) denotes the Cauchy kernel.
  26. [5] Prove that \prod_{i,j,k}(1-x_iy_jz_k)^{-1} = \sum_{\lambda,\mu} (s_\lambda*s_\mu)(x) s_\lambda(y) s_\mu(z), where * denotes the Kronecker product.
  27. Unsolved Let H[a] denote the Heisenberg algebra with parameters a_j (belonging to some ambient field), satisfying [B_k,B_j] = j a_j \delta_{k,-j}. As an abstract algebra, H[a] is isomorphic to the Heisenberg algebra in class, as long as a_j are non-zero. Define U_k by writing h_k in terms of p_\lambda, and replacing p_k by B_{-k}. That is, U_k are the homogeneous symmetric functions inside H[a].
    Let V be a representation of H[a] with distinguished basis {v_\la}, and highest weight vector v_0 (i.e. B_{k}.v_0 = 0 for k > 0). Define formal power series
      G__\la(x_1,x_2,...) = \sum_\alpha x^\alpha [v_\la](U_{\alpha_r}U_{\alpha_{r-1}...U_1.v_0)
    where the summation is over compositions \alpha = (\alpha_1,\alpha_2,...,\alpha_r), and [v_\la] denotes "the coefficient of". Prove that
    (1) G_\la is a symmetric function
    (2) For a suitable action of H[a] on \Lambda, the map \Phi: V -> \Lambda given by \Phi(v_\la) = G_\la is a map of H[a]-modules. (The action is given by scaled multiplication and differentiation by p_k's).
  28. Unsolved A reverse plane partition is a filling F of a Young diagram \la with numbers 1,2,3,... which is weakly increasing along both rows and columns. Define the weight wt(F)=(\alpha_1,\alpha_2,...) by \alpha_i = #{columns containing an i}. Define g_\la = \sum_F x^{wt(F)}, where the summation is over all reverse plane partitions of shape \la. Prove that g_\la is a symmetric function. (Bonus: prove that g_\la is Schur-positive.)
  29. Unsolved Fix k >=2. Define the k-th roots enumerating function f on S_n by f(w) = #{u in S_n | u^k = w}. Prove that f is the character of a representation of S_n.
  30. [8] Find a simple formula for the number of (ordered) pairs u,v \in S_n which commute.
  31. [2] Expand the plethysms h_2[h_n] and h_n[h_2] in terms of Schur functions. Which difference is Schur-positive?
  32. [1] Prove that the degree of the Schubert variety X_\lambda is equal to the number of standard Young tableaux of shape \lambda^c. Here the Grassmannian is embedded in projective space by the map V -> (\Delta_I(V)), called the Plucker embedding.
  33. Unsolved Prove that the Schubert variety X_\lambda is smooth if and only if \lambda^c is a rectangle.
  34. [1]Prove that H^*(Gr(k,n),C) is isomorphic to the quotient of C[e_1,e_2,...,e_k,h_1,h_2,...h_{n-k}] by the ideal generated by the coefficients of the polynomial (1 + e_1 t + e_2 t^2 + ... + e_k t^k) (1 - h_1 t + h_2 t^2 -... + (-1)^{n-k} h_{n-k} t^{n-k}) - 1.
  35. Unsolved Show that \sum_{\lambda,\mu,\nu} (c_{\lambda \mu}^\nu)^2 q^{|\nu|} = \prod_{i >=1} 1/(1-2q^i).
  36. [1] Let w = w_1 w_2 ... w_{2n} be a permutation. Suppose that w_i + w_{2n+1-i} = 2n+1 for all i. Show that the shape of P(w) can be covered by n dominos.
  37. [1] Show that for almost all w \in S_n, the number of bumping operations in RSK applied to w is n^{3/2} (1+o(1)) 128/(27\pi^2) as n goes to infinity.

List of lectures (tentative and potentially ambitious):