Geodesic
Trace identities and central polynomials in the matrix superalgebras $M_{n,k}$
Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 187-206 
Yu. P. Razmyslov. Trace identities and central polynomials in the matrix superalgebras $M_{n,k}$. Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 187-206. http://geodesic.mathdoc.fr/item/SM_1987_56_1_a11/
@article{SM_1987_56_1_a11,
     author = {Yu. P. Razmyslov},
     title = {Trace identities and central polynomials in the matrix superalgebras~$M_{n,k}$},
     journal = {Sbornik. Mathematics},
     pages = {187--206},
     year = {1987},
     volume = {56},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1987_56_1_a11/}
}
TY  - JOUR
AU  - Yu. P. Razmyslov
TI  - Trace identities and central polynomials in the matrix superalgebras $M_{n,k}$
JO  - Sbornik. Mathematics
PY  - 1987
SP  - 187
EP  - 206
VL  - 56
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1987_56_1_a11/
LA  - en
ID  - SM_1987_56_1_a11
ER  - 
%0 Journal Article
%A Yu. P. Razmyslov
%T Trace identities and central polynomials in the matrix superalgebras $M_{n,k}$
%J Sbornik. Mathematics
%D 1987
%P 187-206
%V 56
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1987_56_1_a11/
%G en
%F SM_1987_56_1_a11

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

A complete description is given of trace identities for matrix superalgebras $M_{n,k}=\biggl\{\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\biggr\}$, where $a_{11}$ and $a_{22}$ are square matrices of orders $n$ and $k$ respectively over the even elements of a Grassmann algebra $G$ with countably many generators, while $a_{12}$ and $a_{21}$ are $n\times k$ and $k\times n$ rectangular matrices respectively over the odd elements of $G$. A relation is found between multilinear trace identities of degree $ l$ in the algebra $M_{n,k}$ and irreducible representations of a symmetric group of order $(l+1)!\,$. It is proved that over a field of characteristic zero all trace identities of $M_{n,k}$ follow from identities of degree $nk+n+k$ that hold in that algebra. For every algebra $M_{n,k}$ over a field of arbitrary characteristic a central polynomial is given explicitly. Bibliography: 7 titles.

[1] Razmyslov Yu. P., “Tozhdestva so sledom polnykh matrichnykh algebr nad polem kharakteristiki nul”, Izv. AN SSSR. Ser. matem., 38 (1974), 723–756 | MR | Zbl

[2] Razmyslov Yu. P., “Ob odnoi probleme Kaplanskogo”, Izv. AN SSSR. Ser. matem., 37 (1973), 483–501 | MR | Zbl

[3] Dzheims G., Teoriya predstavlenii simmetricheskikh grupp, Mir, M., 1982 | MR

[4] Kertis Ch., Rainer I., Teoriya predstavlenii konechnykh grupp i assotsiativnykh algebr, Nauka, M., 1969 | MR

[5] Amitsur S. A., “A generalization of Hilbert nullstellensats”, Proc. Amer. Math. Soc., 8:4 (1957), 649–656 | DOI | MR | Zbl

[6] Kon P., Universalnaya algebra, Mir, M., 1968 | MR

[7] Razmyslov Yu. P., “Tozhdestva so sledom superalgebr Mp, k”, XVII Vsesoyuznaya algebraicheskaya konferentsiya, Tezisy soobschenii. ch. 1 (14–17 sentyabrya 1983 g.), Minsk, 1983