 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-103/1/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

For an increasing chain of finite groups

G_{1} \subseteq \dots \subseteq G_{n}

, regard every

C G_{r}

C G_{n}

. The Gelfand–Tsetlin algebra is the unital algebra generated by their centres:

G Z_{n} = alg (Z (C G_{1}), \dots, Z (C G_{n})) .

For this paper the chain is

S_{1} \subseteq \dots \subseteq S_{n}

, so these centres are

Z_{1}, \dots, Z_{n}

. The generators commute: if

r \leq s

, an element of

Z_{s}

commutes with all of

C S_{s}

, including

Z_{r}

. Thus the Gelfand–Tsetlin algebra is commutative. Its joint eigenspaces record successive choices in the restriction of an irreducible representation down the chain.

 New to topics? Read the docs here!