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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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References:
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