# Cocharacters of upper triangular matrices

Document Type : Research Paper

Author

Universita; degli Studi di Bari, II facolta; di scienze, Taranto

Abstract

We survey some recent results on cocharacters of upper triangular matrices‎. ‎In particular‎, ‎we deal both with ordinary and graded cocharacter sequence; we list the principal combinatorial results; we show different techniques in order to solve similar problems‎.

Keywords

Main Subjects

#### References

S. A. Amitsur and A. Regev (1982). PI-algebras and their cocharacters. J. Algebra. 78, 248-254 A. Berele (1982). Homogeneous polynomial identities. Israel J. Math.. 42, 258-272 A. Berele and A. Regev (1998). Codimensions of products and of intersections of verbally prime T-ideals. Israel J. Math.. 103, 17-28 S. Boumova and V. Drensky (2012). Cocharacters of polynomial identities of upper triangular matrices, Article ID 1250018, 24 pp.. J. Algebra Appl.. 11 (1) L. Carini and O. M. Di Vincenzo (1991). On the multiplicities of the cocharacters of the tensor square of the Grassmann algebra. Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur.. 69, 237-246 L. Centrone (2011). Ordinary and $Z_2$-graded cocharacters of $UT_2(E)$. Comm. Algebra. 39 (7), 2554-2572 L. Centrone (2011). Polynomial identities of some minimal varieties with low PI-exponent. PhD thesis. L. Centrone and A. Cirrito (2012). Y-proper graded cocharacters of upper triangular matrices of size 2, 3, 4. J. Algebra. 367, 75-94 O. M. Di Vincenzo (1996). Cocharacters of $G$-graded algebras. Comm. Algebra. 24 (10), 3293-3310 O. M. Di Vincenzo and V. Nardozza (2003). Graded polynomial identities for tensor products by the Grassmann algebra. Comm. Algebra. 31 (3), 1453-1474 O. M. Di Vincenzo, V. Drensky and V. Nardozza (2004). Algebras satisfying the polynomial identity $[x_1,x_2][x_3,x_4,x_5]=0$. J. Algebra Appl.. 3 (2), 121-142 O. M. Di Vincenzo, V. Drensky and V. Nardozza (2003). Subvarieties of the varieties of superalgebras generated by $M_{1,1}(E)$ or $M_2(K)$. Comm. Algebra. 31 (1), 437-461 O. M. Di Vincenzo, P. Koshlukov and A. Valenti (2006). Gradings and graded identities for the upper triangular matrices over an infinite field. Groups, rings and group rings, Lect. Notes Pure Appl. Math., Chapman \& Hall/CRC, Boca Raton, FL. 248, 91-103 V. Drensky (2000). Free algebras and P.I. algebras. Graduate course in algebra, Springer-Verlag Singapore, Singapore. V. Drensky (1988). Extremal varieties of algebras II (Russian). Serdica. 14, 20-27 V. Drensky (1987). Extremal varieties of algebras I (Russian). Serdica. 13, 320-332 V. Drensky (1984). Codimensions of $T$-ideals and Hilbert series of relatively free algebras. J. Algebra. 91, 1-17 V. Drensky (1981). Representations of the symmetric group and varieties of linear algebras. Russian: {em Mat. Sb.}, 115 (1981) 98-115., Translation: {em Math. USSR Sb.}. 43, 85-101 V. Drensky and G. K. Genov (2003). Multiplicities of Schur functions in invariants of two $3\times 3$ matrices. J. Algebra. 264, 496-519 V. Drensky and A. Giambruno (1994). Cocharacters, codimensions and Hilbert series of the polynomial identities for $2\times2$ matrices with involution. Can. J. Math.. 46, 718-733 E. Formanek (1985). Noncommutative invariant theory. Contemp. Math.. 43, 87-119 E. Formanek (1982). Invariants and the ring of generic matrices. Contemp. Math.. 13, 178-223 G. K. Genov (1976). The Spechtness of certain varieties of associative algebras over a field of zero characteristic. Russian: C. R. Acad. Bulgare Sci.. 29, 939-941 G. K. Genov (1981). Some Specht varieties of associative algebras. Russian: Pliska Stud. Math. Bulgare. 2, 30-40 A. Giambruno and M. V. Zaicev Polynomial Identities Algebra and Asymptotic Methods. Mathematical surveys and monographs. 122 A. Giambruno and M. V. Zaicev (2003). Codimension growth and minimal superalgebras. Trans. Amer. Math. Soc.. 355, 5091-5117 A. Giambruno and M. V. Zaicev (2000). A characterization of varieties of associative algebras of exponent two. Serdica Math. J.. 26 (3), 245-252 A. Giambruno and M. V. Zaicev (1999). Exponential codimension growth of P.I. algebras: an exact estimate. Adv. Math.. 142, 221-243 A. Giambruno and M. V. Zaicev (1998). On codimension growth of finitely generated associative algbras. Adv. Math.. 140, 145-155 P. Koshlukov and A. Valenti (2003). Graded identities for the algebra of $n\times n$ upper triangular matrices over an infinite field. Int. J. Algebra Comp.. 13 (5), 517-526 D. Krakovski and A. Regev (1973). The polynomial identities of the Grassmann algebra. Trans. Amer. Math. Soc.. 181, 429-438 V. N. Latyshev (1977). Finite basis property of identities of certain rings. (Russian), Usp. Mat. Nauk. 32(4(196)), 259-260 V. N. Latyshev (1976). Partially ordered sets and nonmatrix identities of associative algebras. Algebra i Logika, 15 (1976) 53--70 (Russian); Translation: Algebra and Logic. 15, 34-45 J. Lewin (1974). A matrix representation for associative algebras I. Trans. Amer. Math. Soc.. 188, 293-308 I. G. Macdonald (1995). Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford, 1979, 2nd edn.. Yu. N. Maltsev (1971). A basis for the identities of the algebra of upper triangular matrices. Algebra i Logika 10 (1971) 393--400, (Russian); Translation: Algebra and Logic. 10, 242-247 S. P. Mishchenko, A. Regev and M. V. Zaicev (1999). A characterization of P.I. algebras with bounded multiplicities of the cocharacters. J. Algebra. 219, 356-368 J. B. Olsson and A. Regev (1976). Colength sequence of some $T$-ideals. J. Algebra. 38, 100-111 A. P. Popov (1982). dentities of the tensor square of a Grassmann algebra. Algebra i Logika, 21 (1982) 442--471 (Russian) Translation: Algebra and Logic. 21, 296-316 Yu. P. Razmyslov (1973). Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero. Algebra i Logika, 12 (1973), 83--113 (Russian) Translation: Algebra and Logic. 12, 47-63 A. Regev (1994). Young-derived sequences of $S_n$-characters. Adv. Math.. 106, 169-197 A. Regev (1979). Algebras satisfying a Capelli identity. Israel J. Math.. 33, 149-154 A. Regev (1972). Existence of identities in $A\otimes B$. Israel J. Math.. 11, 131-152 B. E. Sagan (2000). The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Graduate Texts in Mathematics, Springer-Verlag. 203 A. N. Stoyanova-Venkova (1982). Some lattices of varieties of associative algebras defined by identities of fifth degree. (Russian): C. R. Acad. Bulgare Sci.. 35 (7), 867-868 S. Y. Vasilovsky (1999). $\zn$-graded polynomial identities of the full matrix algebra of order $n$. Proc. Amer. Math. Soc.. 127 (12), 3517-3524