Adam M. Bincer

The Kostant partition function for simple Lie algebras

A simple algorithm is developed for evaluating the Kostant partition function for any simple Lie algebra. The algorithm may also be used to express the partition function for one Lie algebra in terms of the partition function for another, with the latter algebra not necessarily a subalgebra of the former. A special role in the algorithm is played by the sum of the simple roots. Explicit, closed‐form expressions are given for the partition function for a variety of special cases.

Eigenvalues of Casimir operators for the general linear and orthosymplectic Lie superalgebras

A Racah basis is introduced for the generators of these matrix superalgebras and explicit formulas are derived for eigenvalues of Casimir operators in terms of the components of the highest weight. The result contains, as special cases, the corresponding expressions for the general linear, orthogonal, and symplectic Lie algebras.

Missing label operators in the reduction Sp(2n) ↓Sp(2n−2)

I consider the ’’missing label’’ problem for basis vectors of an Sp(2n) representation corresponding to a group reduction chain with links Sp(2ν) ↓Sp(2ν−2) ×U(1), 2⩽ν⩽n. I obtain two different sets of ν−1 missing label operators. These operators, together with the ν Casimir operators of the Sp(2ν) group, the ν−1 Casimir operators of the Sp(2ν−2) subgroup and the one generator of the U(1) subgroup, form a complete set of labeling operators whose eigenfunctions provide a canonical basis in the representation space of Sp(2ν). When the number of missing labels exceeds one the most general solution to the labeling problem is not known. The two particular solutions presented here have certain appealing aspects of symmetry and simplicity.

Normalization coefficients for the O(n) shift operators

Normalization coefficients are derived for the shift operators of O(n) introduced previously. The resultant normalized lowering operator for O(n) is presented in a form analogous to the U(n) case.

Casimir operators of the exceptional group G2

Degree 2 and 6 Casimir operators of G2 are calculated in explicit form, with the generators of G2 written in terms of the subalgebra A2.

Casimir operators for suq(n)

Explicit simple expressions are given for Casimir operators of q-deformed suq(n). These expressions are similar to the corresponding Casimir operators of non-deformed su(n). The proof of invariance is based on direct applications of the commutation relations that define suq(n) written in a convenient basis.

Raising and lowering operators for the orthogonal groups

A formalism for obtaining shift operators for all classical groups in the form of tensor operators is described in application to the orthogonal groups. This formalism is also applicable to the irreps in the discrete series of the noncompact versions of the classical groups.

Mickelsson lowering operators for the symplectic group

Elementary lowering operators for the symplectic group, for which a graphical algorithm was given by Mickelsson, are obtained in the form of tensor operators. The resultant simple analytic expressions are analogous to the corresponding ones found previously for the unitary and orthogonal groups.

Graded contractions of Casimir operators

We describe graded contractions of Casimir operators of Lie algebras. The formalism applies to any Casimir operator given as a symmetric form. We deal with the quadratic Casimir operator in exhaustive detail and indicate the modifications for operators of higher degree.

Missing label operators in the reduction O(p)↓O(p−2)×O(2)

I consider the ‘‘missing label’’ problem for basis vectors of an O(p) representation corresponding to a group reduction chain with links O(p)↓O(p−2)×O(2). A chain with these links is required if the basis vectors are to be of definite weight. I obtain two different sets of missing label operators, which together with the Casimir operators of group and subgroups from a complete set of labeling operators whose eigenvectors provide a canonical basis in the O(p) representation space. The problem is solved for both the even‐ and odd‐dimensional orthogonal groups.