Tensor Methods and a Unified Representation Theory of SU3

Abstract

Starting with irreducible tensors, we develop an explicit construction of orthonormal basic states for an arbitrary unitary irreducible representation (λ, μ) of the group SU3. A knowledge of the simple properties of the irreducible tensors can then be exploited to obtain a variety of results, which ordinarily require more abstract algebraic methods for their derivation. As illustrative applications, we (i) derive Biedenharn's expressions for the matrix elements of the generators of SU3, (ii) compute the matrix elements of octet-type operators for the case (λ, μ) → (λ, μ), and (iii) develop an explicit unitary transformation connecting the isospin and the U-spin states in any arbitrary irreducible representation.

SU(3)-equivariant quiver gauge theories and nonabelian vortices

“…The key to making the above construction explicit is finding a suitable basis for the irreducible representations C k,l of SU (3) that is tailored to the structure of the arrows and vertices of the pertinent quiver, which requires appropriately assembling the invariant (0, 1)-forms into rectangular block matrices. A particularly nice basis for this is the Biedenharn representation [17,18] which takes care of both symmetric and non-symmetric quivers simultaneously. The complete set of d k,l orthonormal basis vectors in this basis set are denoted n q , m and are labelled by the isospin quantum numbers n = 2I, q = 2I z and the hypercharge m = 3Y .…”

“…All of their matrix elements can thus be determined by applying the Wigner-Eckart theorem for SU (2) and relating the resulting reduced matrix elements to an irreducible SU(3) tensor [18]. The latter computation determines numerical coefficient functions…”

“…where the coefficients N 1 (j ± , m ± ), N 2 (j ± , n) and N 3 (n, m) can be found in [18], and the brackets (r, s, t) indicate that the tensor T has r upper or lower indices equal to 1, s equal to 2, and t equal to 3. From the decompositions (2.11), (2.15) and their conjugates it follows that the state T…”

“…Recall that the quantum number q ∈ Z is twice the third component of isospin in the subgroup SU(2) ⊂ SU(3) generated by E ± α 1 , H α 1 , with E α 1 acting by shifts q → q + 2 and leaving the hypercharge quantum number m invariant. The U -spin subgroup of SU (3) is the SU(2) subgroup generated by the operators E ± α 2 , (3) takes isospin into U -spin, and the explicit transformation of states in the Biedenharn basis for C k,l can be found in [18]. The operator E α 2 shifts q U → q U + 2 and leaves m U := 3Y U invariant.…”

Sasakian quiver gauge theories and instantons on cones over lens 5-spaces

“…We decompose C k,l with respect to the subgroup H = SU(2) × U(1) ⊂ G, just as in [23]. A particularly convenient choice of basis for the vector space C k,l is the Biedenharn basis [28][29][30], which is defined to be the eigenvector basis given by…”