Polynomial and Combinatorial Identities Related to Near-Triagular Matrices

A. Guterman

To appear at Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe, Arizona (USA), May 20-26, 2001


We study various combinatorial aspects of concrete algebras satisfying nontrivial polynomial identities (PI-algebras). In particular for the near-triangular matrix algebras we found a basis of their polynomial identities. The general approach to investigate the growth of identities for PI-algebras is to consider their codimension sequences. As of yet the knowledge about codimensions is less than satisfactory. There are only a few concrete algebras with known sequences of codimensions. Here we asymptotically evaluate the growth of identities for near-triangular matrix algebras. It is shown that if an ideal of identities of a certain PI-algebra asymptotically contains all proper identities then this algebra has a polynomial growth of codimensions. An algebra formed by upper triangular matrices with equal elements on the main diagonal is investigated. It is proved that the ideal of identities of this algebra gives an example of ideal which asymptotically contains all proper identities. The codimension sequence for this algebra is computed exactly. As an application of our results the certain combinatorial identity is derived.

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