Bilinear forms on finite abelian groups and group-invariant Butson Hadamard matrices
Let K be a finite abelian group and let exp(K) denote the least common multiple of the orders of the elements of K. A BH(K, h) matrix is a K-invariant |K|×|K| matrix H whose entries are complex hth roots of unity such that HH∗ = |K|I, where H∗ denotes the complex conjugate transpose of H, and I is t...
| Main Authors: | , |
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| Format: | Journal Article |
| Language: | English |
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2020
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| Online Access: | https://hdl.handle.net/10356/143471 |
| _version_ | 1811676771838328832 |
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| author | Duc, Tai Do Schmidt, Bernhard |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Duc, Tai Do Schmidt, Bernhard |
| author_sort | Duc, Tai Do |
| collection | NTU |
| description | Let K be a finite abelian group and let exp(K) denote the least common multiple of the orders of the elements of K. A BH(K, h) matrix is a K-invariant |K|×|K| matrix H whose entries are complex hth roots of unity such that HH∗ = |K|I, where H∗ denotes the complex conjugate transpose of H, and I is the identity matrix of order |K|. Let νp(x) denote the p-adic valuation of the integer x. Using bilinear forms on K, we show that a BH(K, h) exists whenever (i) νp(h) ≥ νp(exp(K))/2 for every prime divisor p of |K| and (ii) ν2(h) ≥ 2 if ν2(|K|) is odd and K has a direct factor Z2. Employing the field descent method, we prove that these conditions are necessary for the existence of a BH(K, h) matrix in the case where K is cyclic of prime power order. |
| first_indexed | 2024-10-01T02:26:46Z |
| format | Journal Article |
| id | ntu-10356/143471 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T02:26:46Z |
| publishDate | 2020 |
| record_format | dspace |
| spelling | ntu-10356/1434712023-02-28T19:21:46Z Bilinear forms on finite abelian groups and group-invariant Butson Hadamard matrices Duc, Tai Do Schmidt, Bernhard School of Physical and Mathematical Sciences Science::Mathematics Group Invariant Matrices Complex Hadamard Matrices Let K be a finite abelian group and let exp(K) denote the least common multiple of the orders of the elements of K. A BH(K, h) matrix is a K-invariant |K|×|K| matrix H whose entries are complex hth roots of unity such that HH∗ = |K|I, where H∗ denotes the complex conjugate transpose of H, and I is the identity matrix of order |K|. Let νp(x) denote the p-adic valuation of the integer x. Using bilinear forms on K, we show that a BH(K, h) exists whenever (i) νp(h) ≥ νp(exp(K))/2 for every prime divisor p of |K| and (ii) ν2(h) ≥ 2 if ν2(|K|) is odd and K has a direct factor Z2. Employing the field descent method, we prove that these conditions are necessary for the existence of a BH(K, h) matrix in the case where K is cyclic of prime power order. Accepted version 2020-09-03T06:22:50Z 2020-09-03T06:22:50Z 2019 Journal Article Duc, T. D., & Schmidt, B. (2019). Bilinear forms on finite abelian groups and group-invariant Butson Hadamard matrices. Journal of Combinatorial Theory, Series A, 166, 337-351. doi:10.1016/j.jcta.2019.03.002 0097-3165 https://hdl.handle.net/10356/143471 10.1016/j.jcta.2019.03.002 2-s2.0-85063758316 166 337 351 en Journal of Combinatorial Theory, Series A © 2019 Elsevier Inc. All rights reserved. This paper was published in Journal of Combinatorial Theory, Series A and is made available with permission of Elsevier Inc. application/pdf |
| spellingShingle | Science::Mathematics Group Invariant Matrices Complex Hadamard Matrices Duc, Tai Do Schmidt, Bernhard Bilinear forms on finite abelian groups and group-invariant Butson Hadamard matrices |
| title | Bilinear forms on finite abelian groups and group-invariant Butson Hadamard matrices |
| title_full | Bilinear forms on finite abelian groups and group-invariant Butson Hadamard matrices |
| title_fullStr | Bilinear forms on finite abelian groups and group-invariant Butson Hadamard matrices |
| title_full_unstemmed | Bilinear forms on finite abelian groups and group-invariant Butson Hadamard matrices |
| title_short | Bilinear forms on finite abelian groups and group-invariant Butson Hadamard matrices |
| title_sort | bilinear forms on finite abelian groups and group invariant butson hadamard matrices |
| topic | Science::Mathematics Group Invariant Matrices Complex Hadamard Matrices |
| url | https://hdl.handle.net/10356/143471 |
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