Separation of Variables for Scalar-valued Polynomials in the Non-stable Range
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913329492000768 |
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| author | Beďatš, Daniel |
| author_facet | Beďatš, Daniel |
| contents | Any complex-valued polynomial on $(\mathbb{R}^n)^k$ decomposes into an algebraic combination of $O(n)$-invariant polynomials and harmonic polynomials. This decomposition, separation of variables, is granted to be unique if $n \geq 2k-1$. We prove that the condition $n\geq 2k-1$ is not only sufficient, but also necessary for uniqueness of the separation. Moreover, we describe the structure of non-uniqueness of the separation in the boundary cases when $n = 2k-2$ and $n=2k-3$.
Formally, we study the kernel of a multiplication map $ϕ$ carrying out separation of variables. We devise a general algorithmic procedure for describing Ker $ϕ$ in the restricted non-stable range $k \leq n < 2k-1$. In the full non-stable range $n < 2k-1$, we give formulas for highest weights of generators of the kernel as well as formulas for its Hilbert series. Using the developed methods, we obtain a list of highest weight vectors generating Ker $ϕ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_11154 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Separation of Variables for Scalar-valued Polynomials in the Non-stable Range Beďatš, Daniel Representation Theory Classical Analysis and ODEs Complex Variables Any complex-valued polynomial on $(\mathbb{R}^n)^k$ decomposes into an algebraic combination of $O(n)$-invariant polynomials and harmonic polynomials. This decomposition, separation of variables, is granted to be unique if $n \geq 2k-1$. We prove that the condition $n\geq 2k-1$ is not only sufficient, but also necessary for uniqueness of the separation. Moreover, we describe the structure of non-uniqueness of the separation in the boundary cases when $n = 2k-2$ and $n=2k-3$. Formally, we study the kernel of a multiplication map $ϕ$ carrying out separation of variables. We devise a general algorithmic procedure for describing Ker $ϕ$ in the restricted non-stable range $k \leq n < 2k-1$. In the full non-stable range $n < 2k-1$, we give formulas for highest weights of generators of the kernel as well as formulas for its Hilbert series. Using the developed methods, we obtain a list of highest weight vectors generating Ker $ϕ$. |
| title | Separation of Variables for Scalar-valued Polynomials in the Non-stable Range |
| topic | Representation Theory Classical Analysis and ODEs Complex Variables |
| url | https://arxiv.org/abs/2309.11154 |