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The dimensions of cyclic symmetry classes of polynomials. (English) Zbl 1290.05156
Let \(H_d[x_1,\dots,x_m]\) be the set of homogeneous polynomials of degree \(d\) over the complex numbers with commuting variables. Let \(G\) be a subgroup of the symmetric group \(S_n\). And let \(\chi\) be a complex, irreducible character of \(G\). Let \(\Gamma^+_{(m,d)}\) be the partitions of \(d\) of length \(m\), and let \(X^\alpha=x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_m^{\alpha_m}\). Then \(\{X^\alpha\mid\alpha\in\Gamma^+_{(m,d)}\}\) is a basis for \(H_d[x_1,\dots,x_m]\).
An inner product for \(H_d\) is given by \(\langle X^\alpha,X^\beta\rangle=\delta_{\alpha,\beta}\). And \(G\) acts on \(H_d[x_1,\dots,x_m]\) by \(q^\sigma(x_1,\dots,x_m)=q(x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(m)})\), where \(q\in H_d\) and \(\sigma\in G\). This action is extended linearly to the group ring of \(G\). Then define the operator \(T\) by \[ T(G,\chi)=\frac{\chi(1)}{|G|}\sum_{\sigma\in G}\chi(\sigma)\sigma. \] The image of \(H_d\) under the action of \(T(G,\chi)\) is the symmetry class of polynomials of degree \(d\) with respect to \(G\) and \(\chi\). \(\{T(G,\chi)\}\) is a complete set of orthogonal idempotents in the group algebra \(\mathbb C(G)\), where \(\chi\) is an irreducible character for \(G\).
If \(\chi\) is a linear character of \(G\), then \(H_d(G,\chi)\) is the set of all \(q\in H_d\) such that \(q^\sigma=\chi(\sigma^{-1})q\). When, for example, \(\chi=1\), the result is the space of homogeneous symmetric polynomials of degree \(d\). And when \(\chi=\epsilon\), one obtains the space of anti-symmetric homogeneous polynomials of degree \(d\).
The paper contains two theorems:
Theorem 3.1 (in which the \(\dim H_d(G,\chi)\) is computed in terms of the generalized Ramanujan sum). Let \(G=C_m\), that is, the cyclic group of order \(m\). Then \[ \dim H_d(G,\chi)=\frac1m\sum_{t|m,m|dt}S(l;m;t)\frac{dt}{m}+t-1{{dt+t-1}\choose{t-1}},\quad t=0,1,\dots,m-1. \] Theorem 3.2. If \(G=C_m\), \((m,d)=1\) and \(\chi\) an irreducible character of \(G\), then
(a)
\(G_\alpha=\{1\}\), for all \(\alpha\in\Gamma^+_{(m,d)}\).
(b)
the set \(\{\sqrt{m}X^{(\alpha,*)}\mid\alpha\in\Delta\}\) is an orthonormal basis of \(H_d(G,\chi)\). In particular,
\[ \dim H_d(G,\chi)=|\Delta|=\frac1m{{dt+m-1}\choose{t-1}}. \]

MSC:
05E05 Symmetric functions and generalizations
15A69 Multilinear algebra, tensor calculus
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