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The Jucys-Murphy elements generate the Gelfand-Tsetlin algebra

Statement

Let $n \geq 1$ . Then $G Z (n) = C [X_{1}, \dots, X_{n}],$ the unital subalgebra generated by the Jucys-Murphy elements; this algebra is the diagonal algebra in the Young basis and is a maximal commutative subalgebra of $C [S_{n}]$ .

Facts & Assumptions

Given: The Gelfand-Tsetlin algebra $G Z (n) = ⟨ Z (C [S_{1}]), \dots, Z (C [S_{n}]) ⟩$ and the Jucys-Murphy elements $X_{1} = 0$ , $X_{k} = \sum_{j < k} (j k)$ for $2 \leq k \leq n$ (The Gelfand-Tsetlin algebra of the symmetric group chain, The Jucys-Murphy elements of the symmetric group algebra, The center $Z (k [G])$ of the group algebra).

[F1]

$G Z (n) = ⨁_{T} C P_{T}$ over the standard tableaux $T$ of size $n$ , where the $P_{T}$ are nonzero pairwise orthogonal idempotents with $\sum_{T} P_{T} = 1$ projecting onto the Young lines; $G Z (n)$ is a maximal commutative subalgebra of $C [S_{n}]$ and is the diagonal algebra in the Young basis (The Gelfand-Tsetlin algebra is the diagonal algebra of the Young basis).

[F2]

Every $P_{T}$ is a polynomial in $X_{1}, \dots, X_{n}$ with rational coefficients, by the interpolation recursion $P_{T} = P_{T ↓ [n - 1]} \prod_{c \in A (μ), c \neq c_{T} (n)} (X_{n} - c) / (c_{T} (n) - c)$ (Primitive tableau idempotents by Jucys-Murphy interpolation).

[F3]

$X_{k} = T_{k} - T_{k - 1}$ for $2 \leq k \leq n$ , where $T_{m} = \sum_{1 \leq i < j \leq m} (i j)$ is the sum of all transpositions of $S_{m}$ ; $T_{m}$ is a class sum, hence central in $C [S_{m}]$ , and $T_{1} = 0$ (The Jucys-Murphy elements of the symmetric group algebra, For a finite group, the class sums form a basis of $Z (k [G])$ ).

Proof

technique · direct

1.1F1F2algebra

Inclusion $G Z (n) \subseteq C [X_{1}, \dots, X_{n}]$ . By [F2] every $P_{T}$ lies in $C [X_{1}, \dots, X_{n}]$ , and by [F1] the $P_{T}$ span $G Z (n)$ ; hence every element of $G Z (n)$ is a polynomial in the $X_{k}$ .

1.2F1F3givenalgebra

Inclusion $C [X_{1}, \dots, X_{n}] \subseteq G Z (n)$ . The element $X_{1} = 0$ lies in $G Z (n)$ ; for $k \geq 2$ , [F3] writes $X_{k} = T_{k} - T_{k - 1}$ as the difference of a central element of $C [S_{k}]$ and a central element of $C [S_{k - 1}]$ , both of which lie in the generating centres of $G Z (n)$ . Since $G Z (n)$ is a subalgebra, it contains every polynomial in the $X_{k}$ .

2.1F1step 1.1step 1.2algebra∎

The two inclusions give $G Z (n) = C [X_{1}, \dots, X_{n}]$ , the algebra generated by the Jucys-Murphy elements; by [F1] this algebra equals $⨁_{T} C P_{T}$ , the diagonal algebra in the Young basis, and is maximal commutative in $C [S_{n}]$ . Finally $C [X_{1}, \dots, X_{n}]$ is reduced and finite-dimensional: it is the algebra of functions on the finitely many content vectors of the standard tableaux, of dimension the number of standard tableaux of size $n$ .

Remarks

Where maximality comes from. The maximal-commutativity assertion is inherited from the diagonal-algebra theorem and not reproved here; the content of the theorem is the equality $G Z (n) = C [X_{1}, \dots, X_{n}]$ , i.e. that the chain of centres and the commuting family of Jucys-Murphy elements generate the same algebra.
No centre-generation input. The inclusion $\subseteq$ uses the interpolation formula for the path idempotents rather than the classical generation of the centre by one-cycle class sums; the inclusion $\supseteq$ uses only $X_{k} = T_{k} - T_{k - 1}$ and the centrality of the transposition sums. No symmetric-function input and no choice principle is used.

Depends on

Used by

Symmetric polynomials in the Jucys-Murphy elements give exactly the centre Theorem

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Okounkov-Vershik, A New Approach to the Representation Theory of the Symmetric Groups, Selecta Math. (N.S.) 2 (1996) 581-605; complete arXiv repost math/0503040, Corollary 2.6 and Proposition 1.1, printed pp. 7-12 (standard reference, not scraped)
Garsia, Young Seminormal Representation, Murphy Elements and Content Evaluations, UCSD lecture notes (2003), Theorems 3.4-3.5, printed pp. 22-24 (standard reference, not scraped)