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Jucys-Murphy element

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Contents

Definition

For nn \in \mathbb{N}, consider the complex group algebra [Sym(n)]\mathbb{C}[Sym(n)] of the symmetric group on nn elements.

Definition

The Jucys-Murphy elements

J k[Sym(n)]k{1,,n} J_k \;\in\; \mathbb{C}[Sym(n)] \;\;\;\; k \in \{1, \cdots, n\}

are the following sums of transpositions (i,j)Sym(n)[Sym(n)](i,j) \in Sym(n) \subset \mathbb{C}[Sym(n)] in this group algebra:

(1)J 10 J_1 \;\coloneqq\; 0
(2)J ki=1k1(i,k) J_k \;\coloneqq\; \underoverset{i = 1}{k-1}{\sum} (i,k)

(Murphy 81, p. 287 (1 of 11), Murphy 83, p. 259 (2 of 8))

Notice that J kJ_k may be understood as the sum of all transpositions in Sym(k)Sym(k) that are not in the subgroup Sym(k1)Sym(k)Sym(k-1) \subset Sym(k).

Properties

Eigenvalues

Write sYTableaux nsYTableaux_n for the set of standard Young tableaux with nn boxes, and write

(3){v T,k[Sym(n)]} TsYTableaux n1mdim(S (|T|)) \big\{ v_{T,k} \in \mathbb{C}[Sym(n)]\big\}_{ { T \in sYTableaux_n} \atop { 1 \leq m \leq dim(S^{(\left\vert T\right\vert)}) } }

for the Gelfand-Tsetlin basis (Young’s seminormal representation).

We will notationally suppress the multiplicity index mm in the following.

Proposition

The v Tv_T (3) are joint eigenvectors of the Jucys-Murphy elements (Def. ) for eigenvalues the “content” jij - i of the box (i,j)(i,j) that contains the number kk in the standard Young tableau

J kv T=(ji)v T,T i,j=k. J_k v_T \;=\; (j - i) v_T \,, \;\;\; T_{i,j} = k \,.

In particular, the Jucys-Murphy elements all commute with each other.

This is due to Jucys 71, recalled as Jucys 74 (12), and independently due to Murphy 81 (3.8).

Factorization of Cayley-Distance kernel

Proposition

The characteristic polynomial of the Jucys-Murphy elements is proportional to the Cayley distance kernel:

(t+J 1)(t+J 2)(t+J n)=σSym(n)e ln(t)#cycles(σ)σ[Sym(n)][t] \big( t + J_1 \big) \big( t + J_2 \big) \cdots \big( t + J_n \big) \;=\; \underset{ \sigma \in Sym(n) }{\sum} e^{ ln(t) \cdot \# cycles(\sigma) } \, \sigma \;\;\;\;\; \in \mathbb{C}[Sym(n)][t]

On the Wikipedia page this is attributed to Jucys, but without explicit reference. It might be in Jucys 71 (but is not mentioned in the review in Jucys 74). However, it is not much of a theorem, anyways:

Proof

By induction:

For n=1n = 1 the claim reduces to

(t+J 1)=e ln(t)1, \big( t + J_1 \big) \;=\; e^{ ln(t) \cdot 1 } \,,

which holds by (1).

Now assume that the statement is true for nn \in \mathbb{N}.

Observe that every permutation σSym(n+1)\sigma \in Sym(n+1) may be written as product

σ=c 1(σ)c 2(σ)c #cycles(σ)(σ) \sigma \;=\; c_1(\sigma) \cdot c_2(\sigma) \cdots c_{\# cycles(\sigma)}(\sigma)

of products of transpositions of the form

c i(σ)=(i i1,i i2)(i 3,i 2)(i 2,i 1)(i 1,i i), c_i(\sigma) \;\; \;=\; (i_{\ell_i-1},i_{\ell_i-2}) \circ \cdots \circ (i_3,i_2) \circ (i_2,i_1) \circ (i_1,i_{\ell_i}) \,,

where i\ell_i is the length of the iith permutation cycle.

Here we may assume without restriction that

i j<i i,AAAandAAAi<ji i<j j i_j \lt i_{\ell_i} \,, {\phantom{AAA}} \text{and} {\phantom{AAA}} i \lt j \;\; \Rightarrow \;\; i_{\ell_i} \lt j_{\ell_j}

because if not then we may (1) rearrange the c i(σ)c_i(\sigma) within their product (these commute with each other, since they contain permutations among elements within distinct cycles) and (2) cyclically rearrange the factors withing each c i(σ)c_i(\sigma).

Once all permutations are uniquely represented as products this way, the representative of any σSym(n+1)\sigma \in Sym(n+1) is uniquely obtained from the representative of an element in Sym(n)Sym(n), by either:

  1. multiplying from the right with a transposition (k,n+1)(k, n+1) for knk \leq n, in which case the n+1n+1st element will be part of a cycle that was already present;

  2. by retaining the previous product as is, in which case the n+1n+1st element will be its own new cycle.

This means that

σSym(n+1)t #cycles(σ)σ=σSym(n)t #cycles(σ)σ(t+J n+1) \underset{\sigma \in Sym(n+1)}{\sum} t^{ \# cycles(\sigma) } \, \sigma \; = \; \underset{\sigma' \in Sym(n)}{\sum} t^{ \# cycles(\sigma') } \, \sigma' (t + J_{n+1})

and hence the claim follows by the induction assumption.

Combining Prop. with Prop. yields:

Corollary

The eigenvalues of the Cayley distance kernel

e ln(t)d C=e ln(t)nσSym(n)e ln(t)#cycles(σ)σ e^{ - ln(t) \cdot d_C } \;=\; e^{- ln(t) \cdot n} \underset{ \sigma \in Sym(n) }{\sum} e^{ ln(t) \cdot \# cycles(\sigma) } \, \sigma \cdot

(regarded on the right as the linear map given by right multiplication in the group algebra) are indexed by Young diagrams λ\lambda and are given by

EigVals[e ln(t)d C] λ=e ln(t)n1irows(λ)1jλ i(tj+i) EigVals[e^{- ln(t) \cdot d_C}]_\lambda \;=\; e^{ - ln(t) \cdot n} \underset{ { 1 \leq i \leq rows(\lambda) } \atop { 1 \leq j \leq \lambda_i } }{\prod} \big( t - j + i \big)

An alternative derivation of this statement, using the formula for Cayley graph spectra and the hook length formula/hook-content formula is this Prop. at Cayley distance kernel (CSS 21).

Projectors

Murphy 81, p. 291 (5 of 11)

Murphy 83, p. 259 (2 of 8)

MathOverflow:a/151375

References

The Jucys-Murphy elements are independently due to:

and

(According to Vershik-Okounkov 04, Footnote 2, Jucys 74 “remained unnoticed” until Murphy 81 first re-discovered the results, and then Jucys’ paper. In fact Jucys 71, which must have the actual proofs, seems to remain electronically unavailable.)

Review:

Further discussion:

  • Kelvin Tian Yi Chan, Induction Relations in the Symmetric Groups and Jucys-Murphy Elements, 2018 (hdl:10012/13601, pdf)

Last revised on May 20, 2021 at 03:25:42. See the history of this page for a list of all contributions to it.