GelfandTsetlin

Gelfand-Tsetlin is a construction in the representation theory of the general linear Lie algebra gl(n) and related groups, named after Israel Gelfand and Michail Tsetlin. It provides an explicit, combinatorial description of finite-dimensional irreducible representations and yields a natural basis for these spaces.

The central objects are Gelfand-Tsetlin patterns, which are triangular arrays of numbers arranged in rows of decreasing length. The entries in adjacent rows satisfy interlacing inequalities, and a pattern encodes a chain of subrepresentations as one restricts from gl(n) to gl(n−1) to gl(n−2), and so on, down to gl(1). Each irreducible representation with a given highest weight carries a basis indexed by such patterns, often called the Gelfand-Tsetlin basis.

A key feature of the construction is the Gelfand-Tsetlin algebra, a maximal commutative subalgebra of the universal enveloping algebra of gl(n) generated by central elements of the subalgebras gl(k) for k = 1, 2, ..., n. This algebra acts diagonally on the GT basis, making the pattern entries natural labels for weight and subalgebra decomposition data. The action of the gl(n) generators on the basis is given explicitly by the Gelfand-Tsetlin formulas, providing concrete raising and lowering operators between basis elements.

Applications of the GT construction include explicit realizations of finite-dimensional representations, combinatorial models for representation-theoretic multiplicities, and connections to harmonic analysis on GL(n) as well as certain quantum integrable systems. It also relates to tableaux combinatorics, with patterns offering an alternative to Young tableaux for indexing basis vectors.