nLab Jucys-Murphy element

Definition

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

Definition

The Jucys-Murphy elements

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

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

(1)$$J_1 \;\coloneqq\; 0$$

(2)$$J_k \;\coloneqq\; \underoverset{i = 1}{k-1}{\sum} (i,k)$$

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

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

Properties

Eigenvalues

Write $sYTableaux_n$ for the set of standard Young tableaux with $n$ boxes, and write

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

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

We will notationally suppress the multiplicity index $m$ in the following.

Proposition

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

$$J_k v_T \;=\; (j - i) v_T \,, \;\;\; T_{i,j} = k \,.$$

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:

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

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 $n = 1$ the claim reduces to

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

which holds by (1).

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

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

$$\sigma \;=\; c_1(\sigma) \cdot c_2(\sigma) \cdots c_{\# cycles(\sigma)}(\sigma)$$

of products of transpositions of the form

$$c_i(\sigma) \;\; \;=\; (i_{\ell_i-1},i_{\ell_i-2}) \circ \cdots \circ (i_3,i_2) \circ (i_2,i_1) \circ (i_1,i_{\ell_i}) \,,$$

where $\ell_i$ is the length of the $i$th permutation cycle.

Here we may assume without restriction that

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

because if not then we may (1) rearrange the $c_i(\sigma)$ 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 $c_i(\sigma)$.

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

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

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

This means that

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

and hence the claim follows by the induction assumption.

Combining Prop. with Prop. yields:

Corollary

The eigenvalues of the Cayley distance kernel

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

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

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

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:

• Algimantas Adolfas Jucys, Factorization of Young projection operators for the symmetric group, Lietuvos Fizikos Rinkinys, 11 (1) 9 (1971) [journal content, pdf]

• Algimantas Adolfas Jucys, Symmetric polynomials and the center of the symmetric group ring, Reports on Mathematical Physics, Volume 5, Issue 1, February 1974, Pages 107-112 (doi:10.1016/0034-4877(74)90019-6)

and

• G. E. Murphy, A new construction of Young’s seminormal representation of the symmetric groups, Journal of Algebra Volume 69, Issue 2, April 1981, Pages 287-297 (doi:10.1016/0021-8693(81)90205-2)

• G. E. Murphy, The idempotents of the symmetric group and Nakayama’s conjecture, Journal of Algebra Volume 81, Issue 1, March 1983, Pages 258-265 (pdf)

• G. E. Murphy, On the representation theory of the symmetric groups and associated Hecke algebras, J. Algebra 152 (1992) 492–513 (pdf, doi:10.1016/0021-8693(92)90045-N))

(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:

• Wikipedia, Jucys-Murphy element

Further discussion:

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