Let O(d) be the special

orthogonal group, acting diagonally on [E.sup.n], and SO(d) the special orthogonal subgroup.

Here the split special

orthogonal group S[O.sub.n] and an embedding S[O.sub.4](F) [??] S[O.sub.5](F) are defined exactly in the same way as [KMS, [section]3].

We write P(m) for the convex cone of real m x m positive definite matrices, and we write @(m) for the

orthogonal group, that is, the set of mxm orthogonal matrices.

The unitary group U(n), the special unitary group SU(n), the

orthogonal group O(n), the special

orthogonal group SO(n), and the symplectic group Sp(n) are the examples of the compact matrix Lie groups [17].

The Schrodinger model for the minimal representation of the indefinite

orthogonal group O(p, q).

A Nested Factors parameterization consisting of g plus three

orthogonal group factors with IT loading on all factors provided a reasonably good fit to the data.

Key words and phrases : cup-length; Lyusternik-Shnirel'man category; rotation group (special

orthogonal group).

Then the homomorphism [rho]: W [right arrow] GL(V) that sends [s.sub.i] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a faithful geometric representation of the Coxeter group W as a discrete subgroup of the

orthogonal group [O.sub.B](V), i.e., the group of linear transformations of V preserving the bilinear form B.

The simple

orthogonal group [O.sub.2k-1] ([F.sub.q]) acts transtively on [OMEGA], and yields a symmetric association scheme [PSI]([O.sub.2k-1]), ([F.sub.q], [OMEGA]) of class (q + 1)/2.

Four lectures and conference talks cover constructing a local conformal field theory associate to a compact Lie group, a level, and a Frobenius object in the corresponding fusion category; field theory interpretation of certain polynomial invariants associated to knots and links; the homotopy theory construction of far-reading generalizations of the topological field theories that Dijkgraf and Witten associated to finite groups; and the action of the

orthogonal group O(n) on the full subcategory of an n-category consisting of the fully dualized objects.

The authors cover the

orthogonal groups, the symmetric case, the short asymmetric case, the Q-tall asymmetric cases, and a wide variety of other related mathematical subjects.

Chapters four through seven cover abstract groups and monoids,

orthogonal groups, stochastic matrices, Lagrange's theorem, groups of units of monoids, homomorphisms, rings, and integral domains.