GelfandTsetlinpatronen

Gelfand-Tsetlin patterns, often abbreviated as GT patterns, are combinatorial objects that arise in the representation theory of the general linear group $GL_n$ and related Lie algebras. They provide a way to index the irreducible finite-dimensional representations of these groups and algebras.

Specifically, GT patterns are arrays of integers that satisfy certain monotonicity and difference conditions. For a representation of $GL_n$, a GT pattern is a triangular array of numbers. Let the pattern be denoted by $\lambda_{i,j}$ for $1 \le i \le j \le n$. The conditions are:

1. $\lambda_{1,1} \ge \lambda_{2,2} \ge \dots \ge \lambda_{n,n}$ (non-increasing along the main diagonal).

2. $\lambda_{i,j} \ge \lambda_{i+1,j}$ for $i < j$ (non-increasing downwards in each column).

3. $\lambda_{i,j} \ge \lambda_{i,j+1}$ for $i < j$ (non-increasing rightwards in each row).

4. $\lambda_{i+1,j+1} \ge \lambda_{i,j}$ for $i < j$ (non-decreasing along diagonals from top-right to bottom-left).

These patterns are in bijection with the extremal weight vectors of irreducible representations of $GL_n$ or the simple Lie algebra $\mathfrak{gl}_n$. The correspondence is established through a recursive procedure involving the weights of the representations.

The study of Gelfand-Tsetlin patterns is significant because they offer an explicit combinatorial description of these representations, which is useful for computing characters, decomposing tensor products, and understanding the structure of the representation theory. They are a fundamental tool in the study of Lie algebras and their representations, particularly in areas related to symmetric functions and combinatorics.