## Planned course on the symmetric group

I’m planning a ten lecture course with the ambitious title `Symmetric functions and symmetric groups’ for next term at RHUL. The audience is likely to be select, but no worse for that. The purpose of this post is to record the proposed syllabus, in particular making sure that my idiosyncratic path is logically sound.

The aim is to emphasise bijective and involutive proofs for symmetric functions, and explicit constructions for the symmetric group. The most obvious omission is the RSK-correspondence. (The various orthogonality relations usually derived from Cauchy’s identity will instead get ad-hoc proofs.)

First lecture: Overview. Frobenius’ characteristic map is an isometric ring isomorphism between the ring of class functions of symmetric groups $\bigoplus_{n=0}^\infty \mathrm{Cl}(S_n)$ and the ring of symmetric functions $\Lambda$. This isomorphism sends the irreducible character $\chi^\lambda$ to the Schur function $S_\lambda$, defined combinatorially by $S_\lambda = \sum_\mu K_{\lambda \mu}m_\mu$, where the Kostka number $K_{\lambda \mu}$ is the number of semistandard $\lambda$-tableaux of type $\mu$. The main aim of the course is to understand this isomorphism. Restricting to $N$ variables, there is a further isomorphism $\Lambda(N) \cong \bigoplus_{n=0}^\infty \mathrm{Rep}_{\mathrm{deg}\;n}(\mathrm{GL}_N(\mathbb{C}))$ sending $S_\lambda(x_1,\ldots,x_N)$ to the character of $\Delta^\lambda(\mathbb{C}^N)$. Examples: the product $s_{(m)} s_{(n)}$ and the plethysm $\mathrm{Sym}^n \mathrm{Sym}^2 (\mathbb{C}^{n})$ interpreted in each setting.

Part A. Symmetric functions.

1. Ring of symmetric functions, monomial, complete, elementary and power-sum symmetric functions. Generating function arguments for Newton type identities. $\omega$ involution defined by $h_n \mapsto e_n$. MacMahon Master Theorem with application to Dixon’s Identity.
2. Bilaternate definition of Schur function $s_\lambda = a_{\lambda + \delta(N)}/a_{\delta(N)}$. The abacus and Loehr’s labelled abacus model for alternating functions. Adding and removing hooks on the abacus. Proof of Pieri’s rule. Young’s rule left as exercise. Corollaries: $\omega(s_\lambda) = s_\lambda'$ and $h_\mu = \sum_\lambda K_{\lambda \mu} s_\lambda$.
3. Proof of Murnaghan–Nakayama rule. Corollary: $p_\mu = \sum_\lambda \theta^\lambda(\mu) s_\lambda$, where $\theta^\lambda(\mu)$ is the sum of the signs of all $\mu$-ribbon tableaux of shape $\lambda$. Corollary: $\omega(p_\mu) = \mathrm{sgn}(\mu) p_\mu$. Brief mention of $\ell$-cores and evaluation of $\chi^\lambda(\ell^m)$ where $\lambda$ is a partition of $\ell m$ with empty $\ell$-core.
4. Lascoux—Schützenberger involution on semistandard tableaux. Corollary: Jacobi–Trudi identity for bialternate Schur functions: $a_{\delta(N)} \sum_{w \in S_N} h_{w \cdot \lambda} = a_{\delta(N) + \lambda}$. Corollary: Littlewood–Richardson rule. (May well be omitted, but it’s not hard from the LS-involution on multitableaux.)
5. Proof of Jacobi—Trudi identity for combinatorial Schur functions $S_\lambda$ using the ant model. Corollary: the two definitions of Schur functions agree.

Part B. Symmetric group.

1. Young permutation modules $M^\mu$ with characters $\pi^\mu$ and Specht modules $S^\lambda$ with characters $\chi^\lambda$. Two row partitions (make connection with Jacobi—Trudi formula) and basic example of a raising map.
2. Young’s rule for the symmetric group, with an unconventional proof. By Frobenius reciprocity, it suffices to show that the submodule of $S^\lambda\downarrow_{S_{n-m} \times S_m}$ on which $S_m$ acts trivially has a Specht filtration by modules $S^\mu \boxtimes F_{S_m}$ where $\mu$ is obtained by removing $m$ boxes from the Young diagram of $\lambda$, no two in the same column. The special case $m=1$ from the elegant and explicitly constructive characteristic-free branching rule proved in Chapter 9 of James’ Springer lecture notes. I will give this proof, and then wave my hands at the generalization, which follows from this paper of James and Peel.
3. Definition of the characteristic map: $\mathrm{ch} \chi^\lambda = s_\lambda$. Definition of inner product on $\Lambda$ so that the Schur functions are an orthonormal basis. Corollary of Young’s rule: $\{ h_\mu \}$ and $\{ m_\mu \}$ are dual bases. Proof that $\sum_\mu \pi^\mu(\lambda) m_\mu = p_\lambda$ and corollary that $\pi^\mu(\lambda) = \langle p_\lambda, h_\mu \rangle$. Corollary: $\mathrm{ch}^{-1} p_\lambda = z_\lambda 1_{\lambda}$, where $1_\lambda$ is the indicator function of elements of $S_n$ of cycle type $\lambda$. Corollary: orthogonality of power-sum symmetric functions. Corollary: Murnaghan—Nakayama rule for symmetric group. Corollary: Littlewood—Richardson rule for symmetric group (only if proved earlier for symmetric functions).

This plans for 9 lectures not 10 so there is a tiny bit of slack, which probably will be needed. If not the final lecture could be on plethysms or other approaches to Young’s rule and the Littlewood—Richardson rule.