A new proof of a theorem of Littlewood
Let \(S_{\mu}(1,q,q^2,\dots )\) denote the ``principal specialization'' of the Schur function \(S_{\mu}(x_1,x_2,\dots )\) of the partition \(\mu\). The coefficient of \(q^k\) in this series gives the multiplicity of the irreducible representation of \(S_n\) indexed by \(\mu\) in the subrepresentation of \(S_n\) on the homogeneous polynomials of degree \(k\). The authors give a new combinatorial proof of the following explicit formula of Littlewood: \[ {S_{\mu}(1,q,q^2,\dots )=\frac{q^{n(\mu)}}{\prod_{s\in\mu}(1-q^{h_{\mu}(s)})}}, \] where \(s\) ranges over the cells of \(\mu\), \(h_{\mu}(s)\) is the hook-length of \(s\) in the diagram of \(\mu\), and \(n(\mu)=\sum_i(i-1)\mu_i\) is the sum of the legs of the cells of \(\mu\). The proof relies on showing that each side of the Littlewood formula satisfies a recursion relation: \[ \frac{q^{n(\mu)}}{\prod_{s\in\mu}(1-q^{h_{\mu}(s)})}B_{\mu}(q,q^{-1})=\frac{1}{1-q}\sum_{\nu\rightarrow\mu}\frac{q^{n(\nu)}}{\prod_{s\in\nu}(1-q^{h_{\nu}(s)})} \] and \[ S_{\mu}(1,q,q^2,\dots )B_{\mu}(q,q^{-1})=\frac{1}{1-q}\sum_{\nu\rightarrow\mu}S_{\nu}(1,q,q^2,\dots ), \] where \(\nu\rightarrow\mu\) means that \(\nu\) immediately precedes \(\mu\) in the Young partial order, and \[ B_{\mu}(q,t)=\frac{1}{qt}\sum_{s\in\mu}q^{(\text{row index of}\,s)}t^{(\text{column index of}\,s)} \] is the biexponent generator of \(\mu\). The first recursion follows easily from a result of \textit{A.M. Garsia} and \textit{G. Tesler} [Adv. Math. 123, No.\,2, 144--222 (1996; Zbl 0865.05075)]. To prove the second recursion, each side is expressed in terms of combinatorial objects: \(S_{\mu}(1,q,q^2,\dots )B_{\mu}(q,q^{-1})=\sum_{n\geq 0}|A_n|q^n\), where \(A_n\) is the set of all pairs \((T,s)\) of semistandard tableaux \(T\) of shape \(\mu\) with entries in \(\mathbb Z_{\geq 0}\), \(s\in\mu\) and \(wt(T)+|s|=n\), and \(\frac{1}{1-q}\sum_{\nu\rightarrow\mu}S_{\nu}(1,q,q^2,\dots )=\sum_{n\geq 0}|B_n|q^n\), where \(B_n\) is the set of all tableaux \(T\) of shape \(\nu\) with entries in \(\mathbb Z_{\geq 0}\), \(\nu\rightarrow\mu\) and \(wt(T)\leq n\). There is a map from \(A_n\) to \(B_n\) given by the process of \textit{jeu de taquin} or \textit{sliding}; the key step is to describe the process of \textit{sliding back} and to show that this is indeed the inverse of the map \(A_n\rightarrow B_n\). The article is clearly written and the bijection between \(A_n\) to \(B_n\) is ilustrated by examples.
scientific article; zbMATH DE number 5523092