# An identity for characters of the symmetric group

I am looking for a reference for the identity $$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$ for the irreducible characters of the symmetric group. It seems quite different from the two results normally seen in textbooks: the Frobenius character formula and the Murnaghan-Nakayama rule. It's a lot easier to prove than either of these, so I am sure it must be known.

Let me first explain the standard notation used in the identity and then indicate how I derived it.

The irreducible character $$\chi_\lambda$$ is labelled by a partition $$\lambda$$ of $$n$$ and $$C$$ is a conjugacy class in $$\mathrm S_n$$. The corresponding representation space $$V_\lambda$$ can be realized as the left ideal $$\mathbb C[\mathrm S_n]c_\lambda$$ in the group algebra. Here, $$c_\lambda=\sum_{p\in P_\lambda,\,q\in Q_\lambda}\operatorname{sgn}(q)pq$$ is the Young symmetrizer. The subgroups $$P_\lambda$$ and $$Q_\lambda$$ in $$\mathrm S_n$$ fix, respectively, the rows and columns in the canonical tableau of shape $$\lambda$$. The dimension $$\dim(V_\lambda)$$ is known from the hook length formula.

The idea is that, up to normalization, $$\chi_\lambda$$ is the intertwining projection of $$c_\lambda$$ onto the space of class functions. For general finite groups, we have two expressions for this projection: $$$$\label{p}(P\phi)(g)=\frac{1}{|G|}\sum_{h\in G}\phi(hgh^{-1})=\sum_{\chi\in\operatorname{Irr}(G)}\langle \phi,\chi\rangle \chi(g),$$$$ where $$\langle \phi,\psi\rangle =\frac{1}{|G|}\sum_{h\in G}\phi(h)\overline{\psi(h)}.$$ I take $$\phi=c_\lambda$$ considered as a function on the group, that is, $$c_\lambda=\sum_g c_\lambda (g)g$$. It follows from well-known facts that $$\langle c_\lambda,\chi_\mu\rangle=\delta_{\lambda\mu}/\dim(V_\lambda)$$, so \begin{align*}\chi_\lambda(g)&=\frac{\dim(V_\lambda)}{n!} \sum_{h\in \mathrm S_n}c_\lambda(h gh^{-1})\\ &=\frac{\dim(V_\lambda)}{n!} \sum_{h\in \mathrm S_n,\,p\in P_\lambda,\,q\in Q_\lambda,\,hpqh^{-1}=g}\operatorname{sgn}(q). \end{align*} Using the stabilizer-orbit theorem applied to the adjoint action, this can be written as above.

• It follows (by taking the trace) from the explicit construction of the Young natural representation in some lecture notes by Garsia. Now it is in a book: Lectures in Algebraic Combinatorics: Young's Construction, Seminormal Representations, SL(2) Representations, Heaps, Basics on Finite Fields (Lecture Notes in Mathematics Book 2277) see link.springer.com/chapter/10.1007/978-3-030-58373-6_1 Commented Dec 18, 2023 at 15:53
• An earlier version is here: garsia.math.yorku.ca/~zabrocki/math6161s14/newIRREPSofSn.pdf The definition of the matrix of the irreducible representation $A^{\lambda}(\sigma)$ is Eq 1.5 in the linked pdf. See also Thm 5.3 Commented Dec 18, 2023 at 16:01
• @AbdelmalekAbdesselam Thanks! I agree that Thm 5.3 in Garsia's notes is equivalent to what I have. This identity must have been known to Young. Commented Dec 18, 2023 at 17:02
• Yes the original reference might be section 18 of Young's QSA3 from 1928. Commented Dec 18, 2023 at 17:45
• I realize that my derivation can be formulated more nicely as follows. The orthogonal projection to the class functions is $Pc_\lambda=(\dim V_\lambda)^{-1} \chi_\lambda$. This means that $\langle c_\lambda,\psi\rangle=(\dim V_\lambda)^{-1}\langle\chi_\lambda,\psi\rangle$ for any class function $\psi$. Now take $\psi$ as the characteristic function of the conjugacy class $C$. This gives $\sum_{g\in C}c_\lambda(g)=(|C|/\dim(V_\lambda))\chi_\lambda(C)$, which is the same identity written in more compact form. Commented Dec 18, 2023 at 19:54