  If a tensor with various symmetries is properly unfolded, then the resulting
matrix inherits those symmetries. As tensor computations become increasingly
important it is imperative that we develop efficient structure preserving
methods for matrices with multiple symmetries. In this paper we consider how to
exploit and preserve structure in the pivoted Cholesky factorization when
approximating a matrix $A$ that is both symmetric ($A=A^T$) and what we call
{\em perfect shuffle symmetric}, or {\em perf-symmetric}. The latter property
means that $A = \Pi A\Pi$ where $\Pi$ is a permutation with the property that
$\Pi v = v$ if $v$ is the vec of a symmetric matrix and $\Pi v = -v$ if $v$ is
the vec of a skew-symmetric matrix. Matrices with this structure can arise when
an order-4 tensor $\cal A$ is unfolded and its elements satisfy ${\cal
A}(i_{1},i_{2},i_{3},i_{4}) = {\cal A}(i_{2},i_{1},i_{3},i_{4}) ={\cal
A}(i_{1},i_{2},i_{4},i_{3}) ={\cal A}(i_{3},i_{4},i_{1},i_{2}).$ This is the
case in certain quantum chemistry applications where the tensor entries are
electronic repulsion integrals. Our technique involves a closed-form block
diagonalization followed by one or two half-sized pivoted Cholesky
factorizations. This framework allows for a lazy evaluation feature that is
important if the entries in $\cal A$ are expensive to compute. In addition to
being a structure preserving rank reduction technique, we find that this
approach for obtaining the Cholesky factorization reduces the work by up to a
factor of 4.
