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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2601.12144 |
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| _version_ | 1866917207665016832 |
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| author | Boumova, Silvia Drensky, Vesselin Fındık, Şehmus |
| author_facet | Boumova, Silvia Drensky, Vesselin Fındık, Şehmus |
| contents | Let $K\langle X_d\rangle$ denote the free associative algebra of rank $d \geq 2$ over a field $K$. By results of Lane (1976) and Kharchenko (1978), the algebra of invariants $K\langle X_d\rangle ^G$ is free for any subgroup $G \leq \GL_d(K)$ and any field $K$.
Koryukin (1984) introduced an additional action of the symmetric group $Sym(n)$ on the homogeneous component of degree $n$ of $K\langle X_d\rangle$, given by permuting the positions of the variables. This endows $K\langle X_d\rangle $ with the structure of a $(K\langle X_d\rangle,\circ)$-$S$-algebra. With respect to this action, Koryukin proved that the invariant algebra $K\langle X_d\rangle ^G$ is finitely generated for every reductive group $G$.
In this paper we study the algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ of invariants under the action of the dihedral group D_{2n} $ on the free associative algebra ${\mathbb C} \langle u,v\rangle$ of rank $2$. We compute the Hilbert series of ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ and construct an explicit set of generators for ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ as a free algebra. Furthermore, we describe a finite generating set for the $S$-algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12144 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On dihedral invariants of the free associative algebra of rank two Boumova, Silvia Drensky, Vesselin Fındık, Şehmus Rings and Algebras Let $K\langle X_d\rangle$ denote the free associative algebra of rank $d \geq 2$ over a field $K$. By results of Lane (1976) and Kharchenko (1978), the algebra of invariants $K\langle X_d\rangle ^G$ is free for any subgroup $G \leq \GL_d(K)$ and any field $K$. Koryukin (1984) introduced an additional action of the symmetric group $Sym(n)$ on the homogeneous component of degree $n$ of $K\langle X_d\rangle$, given by permuting the positions of the variables. This endows $K\langle X_d\rangle $ with the structure of a $(K\langle X_d\rangle,\circ)$-$S$-algebra. With respect to this action, Koryukin proved that the invariant algebra $K\langle X_d\rangle ^G$ is finitely generated for every reductive group $G$. In this paper we study the algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ of invariants under the action of the dihedral group D_{2n} $ on the free associative algebra ${\mathbb C} \langle u,v\rangle$ of rank $2$. We compute the Hilbert series of ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ and construct an explicit set of generators for ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ as a free algebra. Furthermore, we describe a finite generating set for the $S$-algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$. |
| title | On dihedral invariants of the free associative algebra of rank two |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2601.12144 |