The complex conjugate invariants of Clifford groups
- PDF / 315,104 Bytes
- 10 Pages / 439.37 x 666.142 pts Page_size
- 14 Downloads / 171 Views
The complex conjugate invariants of Clifford groups Eiichi Bannai1,2 · Manabu Oura3 · Da Zhao4 Received: 8 June 2020 / Revised: 1 November 2020 / Accepted: 3 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Nebe, Rains and Sloane studied the polynomial invariants for real and complex Clifford groups and they give a new conceptual proof that these invariants are spanned by the set of complete weight enumerators of certain self-dual codes, which was first proved by Runge using modular forms. The purpose of this paper is to show that very similar results can be obtained for the invariants of the complex Clifford group Xm acting on the space of conjugate polynomials in 2m variables of degree N1 in x f and of degree N2 in their complex conjugates x f . In particular, we show that the dimension of this space is 2, for (N1 , N2 ) = (5, 5). This solves affirmatively Conjecture 2 given by Zhu, Kueng, Grassl and Gross. In other words if an orbit of the complex Clifford group is a projective 4-design, then it is automatically a projective 5-design. Keywords Clifford group · Weight enumerator · Self-dual code · Unitary design Mathematics Subject Classification 15A66 · 94B60 · 05B30
Communicated by C. Praeger.
B
Da Zhao [email protected] Eiichi Bannai [email protected] Manabu Oura [email protected]
1
Professor Emeritus of Kyushu University, Fukuoka, Japan
2
Present Address: Asagaya-minami 3-2-33, Suginami-ku, Tokyo 166-0004, Japan
3
Institute of Science and Engineering, Kanazawa University Kakuma-machi, Kanazawa, Ishikawa 920-1192, Japan
4
School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang 200240, Shanghai, China
123
E. Bannai et al.
1 Motivation and background The original motivation of this paper was to settle Conjecture 2 on page 26 of Zhu– Kueng–Grassl–Gross [19]: The Clifford group fails gracefully to be a unitary 4-design, arXiv:1609.08172. The aforementioned conjecture states that if an orbit of the complex Clifford group is a projective 4-design, then it is automatically a projective 5-design. This is equivalent to the statement that a4,4 = a5,5 = 2 in Example 3 in this paper. So the validity of Conjecture 2 was proved. The proof follows step-by-step with some arguments in Nebe–Rains–Sloane’s paper [9]. The aim of design theory is to approximate a space by its finite subsets. There have been numerous studies of designs on intervals, spheres, and Xk , namely the k-subsets of a set X [1, 7]. These designs are useful in areas such as numerical computation and experimental design. Experiments and engineering related to quantum physics raise the need for approximating the space of unitary groups and complex spheres. A unitary t-design is a subset X of the unitary group U (d) such that the average of every function f ∈ Hom(t,t) (U (d)) over X is equal to that over U (d). Here Hom(t,t) (U (d)) is the space of homogeneous complex conjugate polynomials in the entries of the unitary matrix as well as their comp
Data Loading...
