Assignment 4
Due Thursday, March 23, 2023
This assignment is “out of” 100 points, but there are far more than 100 points available. At the instructor’s discretion some “overflow” above 100 may be counted towards your final grade at the end of the course, but you should not expect this.
You are not expected to write up a full solution to every problem, but you are expected to at least think about every problem. Writing up every single problem on the assignment is probably not a good use of your time.
Conventions: Below, \(m_\lambda, e_\lambda, h_\lambda, p_\lambda, s_\lambda\) denote respecticely the monomial, elementary, complete homogeneous, power sum, and Schur bases for the ring of symmetric functions (or symmetric polynomials).
Errata (2023/03/18):
Fixed typo in problem 1.
Errata (2023/03/23):
Fixed error in problem 4.
Let \(p(n)\) denote the number of partitions of the non-negative integer \(n\). Prove that \[\sum_{n=0}^\infty p(n)t^n = \prod_{k=1}^\infty \frac{1}{1-t^k}\] as formal power series (in other words, you do not need to worry about convergence).
Given a partition \(\lambda\), recall that \[z_\lambda = \prod_{i \ge 1} i^{m_i} m_i!\] where \(m_i=m_i(\lambda)\) denotes the multiplicities and \[\epsilon_\lambda = (-1)^{|\lambda|-\ell(\lambda)}\] where \(|\lambda|\) is the weight of \(\lambda\) and \(\ell(\lambda)\) is the length. Let \(\sigma \in S_n\) be a permutation with cycle type \(\lambda\) where \(n = |\lambda|\).
Prove that the centralizer \(Z_{S_n}(\sigma)\) of \(\sigma\) in the group \(S_n\) has order \(z_\lambda\).
Prove that the number of elements of \(S_n\) with cycle type \(\lambda\) is \(n!/z_\lambda\).
Prove \(\epsilon_\lambda = \operatorname{sgn}(\sigma)\) where \(\operatorname{sgn}: S_n \to \{ \pm 1 \}\) is the sign morphism.
Let \(\Delta\) be the Vandermonde determinant in the variables \(x_1,\ldots,x_n\).
Find an explicit expression for \(\Delta^2\) in terms of elementary symmetric polynomials in the case where \(n=2\).
Find an explicit expression for \(\Delta^2\) in terms of elementary symmetric polynomials in the case where \(n=3\).
Prove that \(\sum_{\sigma \in S_n}x^{\sigma(\lambda)} = Cm_\lambda\) where \[C = m_1(\lambda)! \cdots m_r(\lambda)! (n-\ell(\lambda))!\] for a partition \(\lambda\) of length \(r=\ell(\lambda) \le n\).
Prove that \(s_{(d)}=h_d\) and \(s_{(1)^d}=e_d\) for all positive integers \(d\).
Prove that \[h_d = \sum_{\lambda \vdash d} \frac{p_\lambda}{z_\lambda}\] and \[e_d = \sum_{\lambda \vdash d} \epsilon_\lambda\frac{p_\lambda}{z_\lambda}\] for all positive integers \(d\).
Determine the Kostka number \(K_{\lambda \mu}\) for \(\lambda = 41\) and \(\mu = 2111\).
Find explicit expressions for \(p_{111}\), \(p_{21}\), and \(p_3\) in the elementary symmetric function basis.
Recall that the standard representation \(V\) of \(S_n\) is the permutation representation obtained from the action of \(S_n\) on the set \(\{1,\ldots,n\}\). Write \(V\) as a direct sum of Specht modules \(V_\lambda\).
Determine the values of the character \(\chi_\lambda\) of the Specht module \(V_\lambda\) for \(\lambda=32\).
The group \(\operatorname{GL}_n(\mathbb{C})\) acts on the space \(M_n(\mathbb{C})\) of \(n \times n\)-matrices by conjugation. Let \(\mathfrak{sl}_n(\mathbb{C})\) be the subspace of \(M_n(\mathbb{C})\) consisting of matrices with trace \(0\). Prove that \(\mathfrak{sl}_n(\mathbb{C})\) is an irreducible rational subrepresentation. Determine the partition \(\lambda\) and the smallest non-negative integer \(m\) such that \(\mathfrak{sl}_n(\mathbb{C}) \cong \det^m \otimes \mathbb{S}_\lambda(\mathbb{C}^n)\) for a Schur functor \(\mathbb{S}_\lambda\).
Let \(k\) be a field of characteristic \(p > 0\) and let \(V\) be a \(k\)-vector space with basis \(\{x_1,\ldots,x_n\}\). Prove that \[W := \operatorname{span}_k \{ x_1^p, \ldots x_n^p \}\] is a \(\operatorname{GL}(V)\)-stable subspace of the space of symmetric polynomials \(\mathcal{S}^{p}(V)\). (Thus the representation theory of \(\operatorname{GL}(V)\) is very different when the characteristic is not \(0\).)
(Theorem of Clark-Cooper) Let \(f(x)=\sum_{i=0}^n c_ix^{n-i}\) be a monic polynomial over \(\mathbb{C}\) with exactly \(k\) distinct roots \(r_1,\ldots,r_k\). Prove that the values of \(c_1,\ldots,c_k\) and \(r_1,\ldots,r_k\) uniquely determine the multiplicities of the roots. Equivalently, they determine the remaining coefficients \(c_{k+1}, \ldots, c_n\). (Hint: compare the power sums of all roots with the power sums of distinct roots.)