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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2509.01753 |
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| _version_ | 1866908516683350016 |
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| author | Av, Radel Ben Chen, Xuemei Goldberger, Assaf Okoudjou, Kasso A. |
| author_facet | Av, Radel Ben Chen, Xuemei Goldberger, Assaf Okoudjou, Kasso A. |
| contents | This paper studies group frames ($G$-frames) where the unitary group representation can be projective. When the group is abelian, for most combinations $N, n$, we show that $ETF(N,n)$ can only exist for genuinely projective group representations. In particular, cyclic-group frames for such parameters do not exist. We also give a characterization of all dihedral tight frames and dihedral $ETF(2n,n)$, using which, we conclude that regular dihedral $ETF(2n,n)$ must be genuinely projective. Following that, we give a characterization of regular dihedral $ETF(2n,n)$ in terms of certain structured skew Hadamard matrices. We then show that Paley $ETF(2n,n)$ and its doubling are both of this type. Finally, we classify all regular dihedral $ETF(2n,n)$ for $n\le 22$ up to switching equivalence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01753 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Abelian and Dihedral equiangular tight frames of redundancy $2$ Av, Radel Ben Chen, Xuemei Goldberger, Assaf Okoudjou, Kasso A. Combinatorics 05B30, 05E18, 15B34, 22D10 This paper studies group frames ($G$-frames) where the unitary group representation can be projective. When the group is abelian, for most combinations $N, n$, we show that $ETF(N,n)$ can only exist for genuinely projective group representations. In particular, cyclic-group frames for such parameters do not exist. We also give a characterization of all dihedral tight frames and dihedral $ETF(2n,n)$, using which, we conclude that regular dihedral $ETF(2n,n)$ must be genuinely projective. Following that, we give a characterization of regular dihedral $ETF(2n,n)$ in terms of certain structured skew Hadamard matrices. We then show that Paley $ETF(2n,n)$ and its doubling are both of this type. Finally, we classify all regular dihedral $ETF(2n,n)$ for $n\le 22$ up to switching equivalence. |
| title | Abelian and Dihedral equiangular tight frames of redundancy $2$ |
| topic | Combinatorics 05B30, 05E18, 15B34, 22D10 |
| url | https://arxiv.org/abs/2509.01753 |