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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2501.00650 |
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| _version_ | 1866915503930343424 |
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| author | Solomon, David |
| author_facet | Solomon, David |
| contents | We lay the foundations for a broad algebraic theory encompassing SICs in the hope of elucidating their heuristic connections with Stark units. What emerges is a greatly generalised set-up with added structure and potential for applications in other areas. Let $A$ and $B$ be finite modules for a commutative ring $R$, $C$ a finite abelian group and $λ: A\times B\rightarrow C$ an $R$-balanced bilinear pairing. The main constructs are the generalised Heisenberg group ${\cal H}={\cal H}(A,B,C,λ)$ attached to these data (an abstract central extension of $A\oplus B$ by $C$) which plays the role of the Weyl-Heisenberg group in SIC theory, together with its canonical, unitary Schrödinger representations. The SIC itself is replaced by an ${\cal H}$-orbit of complex lines in the representation space, termed a `bouquet'. The overlaps of the SIC are interpreted as a map from ${\cal H}/Z({\cal H})$ into $\mathbb C$ whose absolute values are the Hermitian `angles' between the lines in the bouquet. We also introduce a regularity condition on bouquets in terms of the angle-map, intended to weaken the equiangularity condition of SICs. At the same time, it allows the incorporation of the $R$-structure via the abstract automorphism group of ${\cal H}$ which in turn generalises the Clifford group of SIC theory via its associated Weil representation. As well as several subsidiary definitions and `structural' results, we prove a new `clinometric relation' for the angle-map, determine the structure of the automorphism group and introduce a large class of examples of arithmetic origin, derived from the trace-pairing on quotients of fractional ideals in an arbitrary number field, which we investigate in greater detail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00650 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Towards a Theory of SIC-like Phenomena: Regular Bouquets and Generalised Heisenberg Groups Solomon, David Number Theory 11R04, 20C15, 20D15 (Primary) 11R37, 20C35, 20D45 (Secondary) We lay the foundations for a broad algebraic theory encompassing SICs in the hope of elucidating their heuristic connections with Stark units. What emerges is a greatly generalised set-up with added structure and potential for applications in other areas. Let $A$ and $B$ be finite modules for a commutative ring $R$, $C$ a finite abelian group and $λ: A\times B\rightarrow C$ an $R$-balanced bilinear pairing. The main constructs are the generalised Heisenberg group ${\cal H}={\cal H}(A,B,C,λ)$ attached to these data (an abstract central extension of $A\oplus B$ by $C$) which plays the role of the Weyl-Heisenberg group in SIC theory, together with its canonical, unitary Schrödinger representations. The SIC itself is replaced by an ${\cal H}$-orbit of complex lines in the representation space, termed a `bouquet'. The overlaps of the SIC are interpreted as a map from ${\cal H}/Z({\cal H})$ into $\mathbb C$ whose absolute values are the Hermitian `angles' between the lines in the bouquet. We also introduce a regularity condition on bouquets in terms of the angle-map, intended to weaken the equiangularity condition of SICs. At the same time, it allows the incorporation of the $R$-structure via the abstract automorphism group of ${\cal H}$ which in turn generalises the Clifford group of SIC theory via its associated Weil representation. As well as several subsidiary definitions and `structural' results, we prove a new `clinometric relation' for the angle-map, determine the structure of the automorphism group and introduce a large class of examples of arithmetic origin, derived from the trace-pairing on quotients of fractional ideals in an arbitrary number field, which we investigate in greater detail. |
| title | Towards a Theory of SIC-like Phenomena: Regular Bouquets and Generalised Heisenberg Groups |
| topic | Number Theory 11R04, 20C15, 20D15 (Primary) 11R37, 20C35, 20D45 (Secondary) |
| url | https://arxiv.org/abs/2501.00650 |