Involutions for upper triangular matrix algebras

Involutions and characters of upper triangular matrix groups

We study the realizability over R of representations of the group U(n) of upper-triangular n× n matrices over F2. We prove that all the representations of U(n) are realizable over R if n ≤ 12, but that if n ≥ 13, U(n) has representations not realizable over R. This theorem is a variation on a result that can be obtained by combining work of J. Arregi and A. Vera-López and of the authors, but th...

Involutions on graded matrix algebras

Derived Equivalent Mates of Triangular Matrix Algebras

A triangular matrix algebra over a field k is defined by a triplet (R, S, M) where R and S are k-algebras and RMS is an SR-bimodule. We show that if R, S and M are finite dimensional and the global dimensions of R and S are finite, then the triangular matrix algebra corresponding to (R, S, M) is derived equivalent to the one corresponding to (S, R, DM), where DM = Homk(M, k) is the dual of M , ...

Mixing of the Upper Triangular Matrix Walk

We study a natural random walk over the upper triangular matrices, with entries in the field Z2, generated by steps which add row i + 1 to row i. We show that the mixing time of the lazy random walk is O(n) which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields Zq for q prime.

Algebras and Involutions

• Vectorspaces over division rings • Matrices, opposite rings • Semi-simple modules and rings • Semi-simple algebras • Reduced trace and norm • Other criteria for simplicity • Involutions • Brauer group of a field • Tensor products of fields • Crossed product construction of simple algebras • Cyclic algebra construction of simple algebras • Quaternion algebras • Examples • Unramified extensions...