Hecke Algebras, SVD, and Other Computational Examples with {\sc CLIFFORD}

Abstract

{\sc CLIFFORD} is a Maple package for computations in Clifford algebras $\cl (B)$ of an arbitrary symbolic or numeric bilinear form B. In particular, B may have a non-trivial antisymmetric part. It is well known that the symmetric part g of B determines a unique (up to an isomorphism) Clifford structure on $\cl(B)$ while the antisymmetric part of B changes the multilinear structure of $\cl(B).$ As an example, we verify Helmstetter's formula which relates Clifford product in $\cl(g)$ to the Clifford product in $\cl(B).$ Experimentation with Clifford algebras $\cl(B)$ of a general form~B is highly desirable for physical reasons and can be easily done with {\sc CLIFFORD}. One such application includes a derivation of a representation of Hecke algebras in ideals generated by q-Young operators. Any element (multivector) of $\cl(B)$ is represented in Maple as a multivariate Clifford polynomial in the Grassmann basis monomials although other bases, such as the Clifford basis, may also be used. Using the well-known isomorphism between simple Clifford algebras $\cl(Q)$ of a quadratic form Q and matrix algebras through a faithful spinor representation, one can translate standard matrix algebra problems into the Clifford algebra language. We show how the Singular Value Decomposition of a matrix can be performed in a Clifford algebra. Clifford algebras of a degenerate quadratic form provide a convenient tool with which to study groups of rigid motions in robotics. With the help from {\sc CLIFFORD} we can actually describe all elements of $\Pin(3)$ and $\Spin(3).$ Rotations in $\BR^3$ can then be generated by unit quaternions realized as even elements in $\cl^{+}_{0,3}.$ Throughout this work all symbolic computations are performed with {\sc CLIFFORD} and its extensions.