## 1.1 The Exponent ω of Matrix Multiplication...................... 5 1.2 Groups and Matrix Multiplication.......................... 6 1.2.1 Realizing Matrix Multiplications via Finite Groups............. 6 (709)

### BibTeX

@MISC{7091.1the,

author = {},

title = {1.1 The Exponent ω of Matrix Multiplication...................... 5 1.2 Groups and Matrix Multiplication.......................... 6 1.2.1 Realizing Matrix Multiplications via Finite Groups............. 6},

year = {709}

}

### OpenURL

### Abstract

### Citations

830 | Matrix multiplication via arithmetic progressions - Coppersmith, Winograd - 1990 |

401 | Linear representations of finite groups - Serre - 1977 |

189 |
Algebraic Complexity Theory
- Bürgisser, Clausen, et al.
- 1997
(Show Context)
Citation Context ... interested in worst case complexity.) have proved the following elementary result. (Our focus will be on the Informally, we Proposition 2.11 For every field K, (1) 2 ≤ ω (K) ≤ 3, and (2) ω (K) = h ǫ =-=[2,3]-=- iff Ω(n h+ǫ ) = MK (n) = O ( n h+ε) , where h is uniquely minimal for any given degree of precision ε > 0. The connection between the exponent ω and the concept of bilinear rank is established by the... |

48 | Group-theoretic algorithms for matrix multiplication, preprint
- Cohn, Kleinberg, et al.
(Show Context)
Citation Context ... interested in worst case complexity.) have proved the following elementary result. (Our focus will be on the Informally, we Proposition 2.11 For every field K, (1) 2 ≤ ω (K) ≤ 3, and (2) ω (K) = h ǫ =-=[2,3]-=- iff Ω(n h+ǫ ) = MK (n) = O ( n h+ε) , where h is uniquely minimal for any given degree of precision ε > 0. The connection between the exponent ω and the concept of bilinear rank is established by the... |

36 |
How can we speed up matrix multiplication
- Pan
- 1984
(Show Context)
Citation Context ...ple of [BCS1997, pp. 329-330]. Consider Sym3, the symmetric group of all permutations of 3 symbols. The 6 permutation elements of Sym3 are the identity permutation (1), the three transpositions (12), =-=(13)-=-, and (23), and the two cycles (123) and (132), and these three subsets of elements form the three distinct conjugacy classes Ce, Ct, and Cc, respectively, of D3, where e denotes the no-change permuta... |

33 | A group-theoretic approach to fast matrix multiplication
- Cohn, Umans
- 2003
(Show Context)
Citation Context ...e have proved the following result. Theorem 3.9 For a nontrivial finite group G with a maximal irreducible character degree d ′ (G) and class number c(G) it holds that (1) 1 ≤ (2) 1 = (3) 1 = (4) 1 = =-=(5)-=- 1 < ( ) 1/2 |G| c(G) ( ) |G| 1/2 c(G) ( ) 1/2 |G| c(G) ( ) |G| 1/2 c(G) ( ) 1/2 |G| c(G) ≤ d ′ (G) ≤ (|G| − 1) 1/2 = d ′ (G) ≤ (|G| − 1) 1/2 = d ′ (G) = (|G| − 1) 1/2 = d ′ (G) < (|G| − 1) 1/2 < d ′ ... |

29 | Gaussian Elimination is not - Strassen - 1969 |

16 | Character theory of finite groups, Walter de Gruyter - Huppert - 1998 |

16 | Geometry and the complexity of matrix multiplication - Landsberg |

15 | The Cooley{Tukey FFT and group theory - Maslen, Rockmore |

15 | Asymptotics of the largest and the typical dimensions of irreducible representations of a symmetric group, Funktsional. Anal. i Prilozhen - Vershik, Kerov - 1985 |

6 | Group-theoretic Algorithms for Matrix Multiplication’, (private email - Umans |

2 |
Berekeningscomplexiteit van Bilineaire en Kwadratische Vormen (Computational Complexity of Bilinear and Quadratic Forms
- Boas
- 1982
(Show Context)
Citation Context ... estimates for ω can be derived from certain numerical parameters relating to the efficiency with which groups realize matrix multiplications, and also to the degrees of their irreducible characters. =-=(1)-=- Pseudoexponents α(G), defined by α(G) := log z ′ (G) 1/3 |G|, where tensor of maximal size z ′ (G) = n ′ m ′ p ′ 〈 n ′ ,m ′ ,p ′〉 is a matrix > 1 realized by G, and uniquely determining α(G). We prov... |