Geodesic
The isotropy group of the matrix multiplication tensor
Trudy Instituta matematiki, Tome 24 (2016) no. 2, pp. 106-118.

Voir la notice de l'article provenant de la source Math-Net.Ru

By an isotropy group of a tensor $t\in V_1 \otimes V_2\otimes V_3=\widetilde V$ we mean the group of all invertible linear transformations of $\widetilde V$ that leave $t$ invariant and are compatible (in an obvious sense) with the structure of tensor product on $\widetilde V.$ We consider the case where $t$ is the structure tensor of multiplication map of rectangular matrices. The isotropy group of this tensor was studied in 1970s by de Groote, Strassen, and Brockett-Dobkin. In the present work we enlarge, make more precise, expose in the language of group actions on tensor spaces, and endow with proofs the results previously known. This is necessary for studying the algorithms of fast matrix multiplication admitting symmetries. The latter seems to be a promising new way for constructing fast algorithms. 
@article{TIMB_2016_24_2_a10,
     author = {V. P. Burichenko},
     title = {The isotropy group of the matrix multiplication tensor},
     journal = {Trudy Instituta matematiki},
     pages = {106--118},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2016_24_2_a10/}
}
TY  - JOUR
AU  - V. P. Burichenko
TI  - The isotropy group of the matrix multiplication tensor
JO  - Trudy Instituta matematiki
PY  - 2016
SP  - 106
EP  - 118
VL  - 24
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TIMB_2016_24_2_a10/
LA  - en
ID  - TIMB_2016_24_2_a10
ER  - 
%0 Journal Article
%A V. P. Burichenko
%T The isotropy group of the matrix multiplication tensor
%J Trudy Instituta matematiki
%D 2016
%P 106-118
%V 24
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TIMB_2016_24_2_a10/
%G en
%F TIMB_2016_24_2_a10
V. P. Burichenko. The isotropy group of the matrix multiplication tensor. Trudy Instituta matematiki, Tome 24 (2016) no. 2, pp. 106-118. http://geodesic.mathdoc.fr/item/TIMB_2016_24_2_a10/

[1] Burichenko V. P., On symmetries of the Strassen algorithm, 2014, arXiv: 1408.6273 | Zbl

[2] Burichenko V. P., Symmetries of matrix multiplication algorithms. I, 2015, arXiv: 1508.01110 | Zbl

[3] Strassen V., “Gaussian elimination is not optimal”, Numer. Math., 13 (1969), 354–356 | DOI | MR | Zbl

[4] Hopcroft J. E., Kerr L. R., “On minimizing the number of multiplications necessary for matrix multiplication”, SIAM J.Appl.Math., 20 (1971), 30–36 | DOI | MR | Zbl

[5] Laderman J., “A noncommutative algorithm for multiplying $3\times3$ matrices using 23 multiplications”, Bull. Amer. Math. Soc., 82 (1976), 126–128 | DOI | MR | Zbl

[6] Burichenko V. P., Symmetries of matrix multiplication algorithms, In preparation

[7] de Groote H. F., “On the varieties of optimal algorithms for the computation of bilinear mappings. I. The isotropy group of a bilinear mapping”, Theor. Comput. Sci., 7 (1978), 1–24 | DOI | MR | Zbl

[8] de Groote H. F., “On the varieties of optimal algorithms for the computation of bilinear mappings. II. Optimal algorithms for $2\times2$ matrix multiplication”, Theor. Comput. Sci., 7 (1978), 127–148 | DOI | MR | Zbl

[9] Brockett R. W., Dobkin D., “On the optimal evaluation of a set of bilinear forms”, Linear Algebra Appl., 19 (1978), 207–235 | DOI | MR | Zbl

[10] Chiantini L., Ikenmeyer C., Landsberg J. M., Ottaviani G., The geometry of rank decompositions of matrix multiplication I: $2\times2$ matrices, arXiv: 1610.08364v1

[11] Kostrikin A. I., Manin Yu.I., Linear Algebra and Geometry, 2nd ed., Gordon and Breach, 1997 | MR | Zbl

[12] Kostrikin A. I., Introduction to Algebra, Nauka, M., 1977 (in Russian) | MR | Zbl

[13] Curtis C. W., Reiner I., Representation Theory of Finite Groups and Associative Algebras, Interscience Publishers, 1962 | MR | Zbl

[14] Burau W., Mehrdimensionale Projective und Hohere Geometrie, Deutsche Verlag der Wissenschaften, Berlin, 1961 | MR

[15] Grochow J. A., Moore C., Matrix multiplication algorithms from group orbits, arXiv: 1612.01527v1