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**1 - 6**of**6**### Results for the Exponent ω of Matrix Multiplication

, 2007

"... This review paper describes certain elementary methods of group theory for studying the algebraic complexity of matrix multiplication, as measured by the exponent 2 ≤ ω ≤ 3 of matrix multiplication. which is conjectured to be 2. The seed of these methods lies in two ideas of H. Cohn, C. Umans et. al ..."

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This review paper describes certain elementary methods of group theory for studying the algebraic complexity of matrix multiplication, as measured by the exponent 2 ≤ ω ≤ 3 of matrix multiplication. which is conjectured to be 2. The seed of these methods lies in two ideas of H. Cohn, C. Umans et. al., firstly, that it is possible to ”realize ” a matrix product via the regular algebra of a (finite) group having a triple of subsets satisfying the so-called triple product property (TPP), and more generally, that it is possible to simultaneously realize several independent matrix products via a single group having a family of triples of subsets satisfying the so-called simultaneous triple product property (STPP), in such a way that the complexity of these several multiplications does not exceed the complexity of one multiplication in the regular algebra of the group. The STPP, in particular, has certain implications for ω which we describe. The most general result which we’ve obtained is that if an Abelian group H simultaneously realizes n matrix products of dimensions mi × pi by pi × qi then the wreath product group H ≀ Symn realizes some 1 ≤ k = k (n) < (n!) 3 matrix products of equal dimensions n! nQ mi × n! nQ pi by n! nQ pi × n! nQ n log|H|−log n!−log k qi such that ω ≤ v. i=1 i=1 i=1 i=1 log 3 u t nQ mipiqi i=1 As an application of this result, we prove that ω ≤ 2n log n 3n −log 2 n!−log k 2nn log(n−1) for some 1 ≤ k < (2 n!) 3 using the Abelian group ` Cyc ×3´×n n and its wreath product ` Cyc ×3´×n n ≀ Sym2n. The sharpest estimate for ω using these groups occurs for n = 25 when ω < 225 log 25 75 −log 2 25!−log k

### Results for the Exponent ω of Matrix Multiplication

, 2007

"... This review paper describes certain elementary methods of group theory for studying the algebraic complexity of matrix multiplication, as measured by the exponent 2 ≤ ω ≤ 3 of matrix multiplication. which is conjectured to be 2. The seed of these methods lies in two ideas of H. Cohn, C. Umans et. al ..."

Abstract
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This review paper describes certain elementary methods of group theory for studying the algebraic complexity of matrix multiplication, as measured by the exponent 2 ≤ ω ≤ 3 of matrix multiplication. which is conjectured to be 2. The seed of these methods lies in two ideas of H. Cohn, C. Umans et. al., firstly, that it is possible to ”realize ” a matrix product via the regular algebra of a (finite) group having a triple of subsets satisfying the so-called triple product property (TPP), and more generally, that it is possible to simultaneously realize several independent matrix products via a single group having a family of triples of subsets satisfying the so-called simultaneous triple product property (STPP), in such a way that the complexity of these several multiplications does not exceed the complexity of one multiplication in the regular algebra of the group. The STPP, in particular, has certain implications for ω which we describe. The most general result which we’ve obtained is that if an Abelian group H simultaneously realizes n matrix products of dimensions mi × pi by pi × qi then the wreath product group H ≀ Symn realizes some 1 ≤ k = k (n) < (n!) 3 matrix products of equal dimensions n! nQ mi × n! nQ pi by n! nQ pi × n! nQ n log|H|−log n!−log k qi such that ω ≤ v. i=1 i=1 i=1 i=1 log 3 u t nQ mipiqi i=1 As an application of this result, we prove that ω ≤ 2n log n 3n −log 2 n!−log k 2nn log(n−1) for some 1 ≤ k < (2 n!) 3 using the Abelian group ` Cyc ×3´×n n and its wreath product ` Cyc ×3´×n n ≀ Sym2n. The sharpest estimate for ω using these groups occurs for n = 25 when ω < 225 log 25 75 −log 2 25!−log k

### Multiplication

, 2007

"... We derive general bounds and specific estimates for the exponent 2 ≤ ω ≤ 3 of matrix multiplication, which is conjectured to be 2, by using elementary methods from group theory to ”realize ” simultaneous matrix multiplications via the regular group algebras of finite groups. Our main result is that ..."

Abstract
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We derive general bounds and specific estimates for the exponent 2 ≤ ω ≤ 3 of matrix multiplication, which is conjectured to be 2, by using elementary methods from group theory to ”realize ” simultaneous matrix multiplications via the regular group algebras of finite groups. Our main result is that ω < 225 log 25 75 −log 2 25!−log k

### Submitted: 31/03/2007

, 2007

"... We derive general bounds and specific estimates for the exponent 2≤ ω ≤ 3 of matrix multiplication, which is conjectured to be 2, by using elementary methods from group theory to ”realize ” simultaneous matrix multiplications via the regular group algebras of finite groups. Our main result is that ω ..."

Abstract
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We derive general bounds and specific estimates for the exponent 2≤ ω ≤ 3 of matrix multiplication, which is conjectured to be 2, by using elementary methods from group theory to ”realize ” simultaneous matrix multiplications via the regular group algebras of finite groups. Our main result is that ω < 225 log 25 75 −log 2 25!−log k 2 25 25 log 24 < 2.84 for some 1≤k< ` 2 25! ´ 3 This review paper describes the nature and applications of certain elementary methods of group theory for studying the algebraic complexity of matrix multiplication, as measured by the exponent ω of matrix multiplication, as developed by H. Cohn, C. Umans et. al, (see [CU2003], [CU2005]). These are methods for studying ω by ”realizing ” matrix multiplications via embeddings in the regular group algebras of finite groups having appropriate ”index ” subsets, or families of such subsets, satisfying the so-called triple product, and simultaneous triple product, properties. We pursue the latter, in particular, which yields several types of general estimates for ω involving the sizes of the embedding groups and their index subsets, as well as the dimensions of their irreducible representations. These general estimates also lead to specific estimates for ω when we work with specific types of groups, especially Abelian groups, such as Cyc ×3 n, and wreath product groups H ≀Symn, where H is Abelian. Using these results, our main original result is that ω < 225 log 25 75 −log 2 25!−log k