theory of dual pairs. The special case where σ is the trivial representation of H was first used extensively by Hua in his work on the Szegő kernels of bounded symmetric domains in several complex variables, where the Shilov boundary has the form G/H. More generally the Cartan-Helgason theorem gives the decomposition when G/H is a compact symmetric space, in which case all multiplicities are one; a generalization to arbitrary σ has since been obtained by . Similar geometric considerations have also been used by to rederive Littlewood's rules, which involve the celebrated Littlewood–Richardson rules for tensoring irreducible representations of the unitary groups. has found generalizations of these rules to arbitrary compact semisimple Lie groups, using his path model, an approach to representation theory close in spirit to the theory of crystal bases of Lusztig and Kashiwara. His methods yield branching rules for restrictions to subgroups containing a maximal torus. The study of branching rules is important in classical invariant theory and its modern counterpart, algebraic combinatorics. Example. The unitary group U(N) has irreducible representations labelled by signatures where the fi are integers. In fact if a unitary matrix U has eigenvalues zi, then the character of the corresponding irreducible representation f is given by The branching rule from U(N) to U(N – 1) states that {| border="1" cellspacing="0" cellpadding="5" | |} Example. The unitary symplectic group or quaternionic unitary group, denoted Sp(N) or U(N, H), is the group of all transformations of HN which commute with right multiplication by the quaternions H and preserve the H-valued hermitian inner product on HN, where q* denotes the quaternion conjugate to q. Realizing quaternions as 2 x 2 complex matrices, the group Sp(N) is just the group of block matrices (qij) in SU(2N) with where αij and βij are complex numbers. Each matrix U in Sp(N) is conjugate to a block diagonal matrix with entries where |zi| = 1. Thus the eigenvalues of U are (zi±1). The irreducible representations of Sp(N) are labelled by signatures where the fi are integers. The character of the corresponding irreducible representation σf is given by The branching rule from Sp(N) to Sp(N – 1) states that {| border="1" cellspacing="0" cellpadding="5" | |} Here fN + 1 = 0 and the multiplicity m(f, g) is given by where is the non-increasing rearrangement of the 2N non-negative integers (fi), (gj) and 0. Example. The branching from U(2N) to