A note on Cartan matrices for symmetric groups.
Let \(n\) and \(e\) be positive integers with \(e\geq 2\) and let \(\lambda\) be a partition of \(n\) (\(\lambda\vdash n\)) and write \(\lambda=(1^{a_1(\lambda)},2^{a_2(\lambda)},\dots)\) where \(a_i(\lambda)\) is the multiplicity of \(i\) as a part of the partition \(\lambda\). Also let \(\ell(\lambda)\) denote the length (number of parts) of \(\lambda\). Let \(T(q)\), \(P(q)\) and \(L(q)\) denote the generating functions for \(d(n)=\) the number of divisors of \(n\), \(p(n)=\) the number of partitions of \(n\) and \(\ell(n)=\sum_{\lambda\vdash n}\ell(\lambda)\), resp. A partition \(\lambda\) of \(n\) is called \(e\)-class regular if \(a_i(\lambda)=0\) whenever \(e\mid i\) and in that case, write \(\lambda\vdash_en\). Let \(T_e(q)\) be the generating function for the number of divisors of \(n\) that are not divisible by \(e\). Let \(p_e(n)\) be the number of \(e\)-class regular partitions of \(n\), let \(\ell_e(n)=\sum_{\lambda\vdash_en}\ell(\lambda)\) and let \(P_e(q)\) and \(L_e(q)\) denote the corresponding generating functions, resp. Sections 1 and 2 present various relations among the above generating functions. For an \(e\)-class regular partition \(\lambda\) of \(n\), define the \(e\)-defect of \(\lambda\) by: \(d_e(\lambda)=\sum_{i,j\geq 1}[\tfrac{a_i(\lambda)}{ej}]\) (where \([\cdots]\) denotes ``the integral part of'') and set \(c_e(n)=\sum_{\lambda\vdash_en}d_e(\lambda)\). A partition \(\lambda\vdash_en\) is said to be of \(e\)-class defect \(0\) if \(d_e(\lambda)=0\) (this is the case if and only if \(a_i(\lambda)\leq e-1\) for all \(i\)). The number of such partitions is denoted by \(d^0_e(n)\) and let \(D^0_e(q)\) denote the corresponding generating function. The first main result of this paper is: Theorem 3.3. \(C_e(q)=P_e(q)T(q^e)\). In analogy with \(p\)-modular representations of \(S_n\), let \(\overline d_e(n)\) denote the number of \(e\)-core partitions of \(n\), so that the generating function for the numbers \(\overline d_e(n)\) is \(\overline D_e(q)=P(q)/P(q^e)^e\). Analogously define \(\overline c_e(n)\). The final main result of this paper is: Theorem 3.4. Let \(\overline C_e(q)\) be the generating function for \(\overline c_e(n)\). Then \(\overline C_e(q)=P(q)^{e-1}T(q)\). The motivation for and connections of these results with the \(p\)-modular representation theory of the symmetric groups is fully explained in this article. A discussion of some recent developments in this area is also presented here.
scientific article; zbMATH DE number 2051629