A Positivstellensatz on the Matrix Algebra of Finitely Generated Free Group
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913386597449728 |
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| author | Liang, Hao |
| author_facet | Liang, Hao |
| contents | Positivstellens{ä}tze are a group of theorems on the positivity of involution algebras over $\mathbb{R}$ or $\mathbb{C}$. One of the most well-known Positivstellensatz is the solution to Hilbert's 17th problem given by E. Artin, which asserts that a real polynomial in $n$ commutative variables is nonnegative on real affine space if and only if it is a sum of fractional squares. Let $m$ and $n$ be two positive integers. For the free group $F_n$ generated by $n$ letters, and a symmetric polynomial $b$ with variables in $F_n$ and with $n$-by-$n$ complex matrices coefficients, we use real algebraic geometry to give a new proof showing that $b$ is a sum of Hermitian squares if and only if $b$ is mapped to a positive semidefinite matrix under any finitely dimensional unitary representation of $F_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07367 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Positivstellensatz on the Matrix Algebra of Finitely Generated Free Group Liang, Hao Representation Theory Group Theory Positivstellens{ä}tze are a group of theorems on the positivity of involution algebras over $\mathbb{R}$ or $\mathbb{C}$. One of the most well-known Positivstellensatz is the solution to Hilbert's 17th problem given by E. Artin, which asserts that a real polynomial in $n$ commutative variables is nonnegative on real affine space if and only if it is a sum of fractional squares. Let $m$ and $n$ be two positive integers. For the free group $F_n$ generated by $n$ letters, and a symmetric polynomial $b$ with variables in $F_n$ and with $n$-by-$n$ complex matrices coefficients, we use real algebraic geometry to give a new proof showing that $b$ is a sum of Hermitian squares if and only if $b$ is mapped to a positive semidefinite matrix under any finitely dimensional unitary representation of $F_n$. |
| title | A Positivstellensatz on the Matrix Algebra of Finitely Generated Free Group |
| topic | Representation Theory Group Theory |
| url | https://arxiv.org/abs/2406.07367 |