On the multiplicities of the central cocharacter of algebras with polynomial identities
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| Format: | Preprint |
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2026
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| _version_ | 1866909990389809152 |
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| author | Cota, Wesley Quaresma Nascimento, Thais Silva do |
| author_facet | Cota, Wesley Quaresma Nascimento, Thais Silva do |
| contents | For an associative algebra $A$ over a field of characteristic zero, let $P_n(A)$ and $P_n^z(A)$ denote the spaces of multilinear polynomials of degree $n$ modulo the polynomial identities and the central polynomials of $A$, respectively. We also write $Δ_n(A)$ for the space of multilinear central polynomials of degree $n$ modulo the polynomial identities of $A$. The corresponding sequences of colengths, central colengths and proper central colengths measure the number of irreducible components in the $S_n$-module decompositions of $P_n(A)$, $P_n^z(A)$ and $Δ_n(A)$, respectively. In this paper, we investigate several examples of PI-algebras and explicitly describe their cocharacter, central cocharacter and proper central cocharacter sequences. As a consequence, we obtain a complete classification, up to PI-equivalence, of all algebras whose sequences of colengths and central colengths are bounded by a constant. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_09546 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the multiplicities of the central cocharacter of algebras with polynomial identities Cota, Wesley Quaresma Nascimento, Thais Silva do Rings and Algebras Primary 16R10, 16W50, Secondary 20C30 For an associative algebra $A$ over a field of characteristic zero, let $P_n(A)$ and $P_n^z(A)$ denote the spaces of multilinear polynomials of degree $n$ modulo the polynomial identities and the central polynomials of $A$, respectively. We also write $Δ_n(A)$ for the space of multilinear central polynomials of degree $n$ modulo the polynomial identities of $A$. The corresponding sequences of colengths, central colengths and proper central colengths measure the number of irreducible components in the $S_n$-module decompositions of $P_n(A)$, $P_n^z(A)$ and $Δ_n(A)$, respectively. In this paper, we investigate several examples of PI-algebras and explicitly describe their cocharacter, central cocharacter and proper central cocharacter sequences. As a consequence, we obtain a complete classification, up to PI-equivalence, of all algebras whose sequences of colengths and central colengths are bounded by a constant. |
| title | On the multiplicities of the central cocharacter of algebras with polynomial identities |
| topic | Rings and Algebras Primary 16R10, 16W50, Secondary 20C30 |
| url | https://arxiv.org/abs/2601.09546 |