Jucys–Murphy element

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title: "Jucys–Murphy element" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["permutation-groups", "representation-theory", "symmetry", "representation-theory-of-finite-groups", "symmetric-functions"] description: "Elements in representations of the symmetric group" topic_path: "general/permutation-groups" source: "https://en.wikipedia.org/wiki/Jucys–Murphy_element" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Elements in representations of the symmetric group ::

In mathematics, the Jucys–Murphy elements in the group algebra \mathbb{C} [S_n] of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula:

:X_1=0, ~~~ X_k= (1 ; k)+ (2 ; k)+\cdots+(k-1 ; k), ~~~ k=2,\dots,n.

They play an important role in the representation theory of the symmetric group.

Properties

They generate a commutative subalgebra of \mathbb{C} [ S_n] . Moreover, X**n commutes with all elements of \mathbb{C} [S_{n-1}] .

The vectors constituting the basis of Young's "seminormal representation" are eigenvectors for the action of X**n. For any standard Young tableau U we have:

:X_k v_U =c_k(U) v_U, ~~~ k=1,\dots,n,

where c**k(U) is the content b − a of the cell (a, b) occupied by k in the standard Young tableau U.

Theorem (Jucys): The center Z(\mathbb{C} [S_n]) of the group algebra \mathbb{C} [S_n] of the symmetric group is generated by the symmetric polynomials in the elements Xk.

Theorem (Jucys): Let t be a formal variable commuting with everything, then the following identity for polynomials in variable t with values in the group algebra \mathbb{C} [S_n] holds true:

: (t+X_1) (t+X_2) \cdots (t+X_n)= \sum_{\sigma \in S_n} \sigma t^{\text{number of cycles of }\sigma}.

Theorem (Okounkov–Vershik): The subalgebra of \mathbb{C} [S_n] generated by the centers

: Z(\mathbb{C} [ S_1]), Z(\mathbb{C} [ S_2]), \ldots, Z(\mathbb{C} [ S_{n-1}]), Z(\mathbb{C} [S_n])

is exactly the subalgebra generated by the Jucys–Murphy elements Xk.

References

• {{Citation |title=A New Approach to the Representation Theory of the Symmetric Groups. 2 |authorlink1=Okounkov |first1=Andrei |last1=Okounkov |authorlink2=Vershik |first2=Anatoly |last2=Vershik |year=2004 |volume=307 |journal=Zapiski Seminarov POMI |arxiv = math.RT/0503040 |postscript= (revised English version). }}

• {{citation |title=Symmetric polynomials and the center of the symmetric group ring |authorlink1=Algimantas Adolfas Jucys |first1=Algimantas Adolfas |last1=Jucys | year=1974 | journal=Rep. Mathematical Phys. | volume=5 | issue=1 | pages=107–112 | doi=10.1016/0034-4877(74)90019-6 |bibcode=1974RpMP....5..107J}}

• {{citation |title=On the Young operators of the symmetric group |authorlink1=Algimantas Adolfas Jucys |first1=Algimantas Adolfas |last1=Jucys | year=1966 | journal=Lietuvos Fizikos Rinkinys | volume=6 | pages=163–180 |url=https://www.lietuvos-fizikai.lt/chessidr/straipsniai/LietFizRink/LFR-1966-v6-p179-AlgJucys-On_the_Young_operators_of_the_symmetric_groups.pdf

• {{citation |title=Factorization of Young projection operators for the symmetric group |authorlink1=Algimantas Adolfas Jucys |first1=Algimantas Adolfas |last1=Jucys | year=1971 | journal=Lietuvos Fizikos Rinkinys | volume=11 | pages=5–10 |url=https://www.lietuvos-fizikai.lt/chessidr/straipsniai/LietFizRink/LFR-1971-v11-p5-AlgJucys-Factorization_of_Young_projection_operators_for_the_symmetric_group.pdf

• {{citation |title=A new construction of Young's seminormal representation of the symmetric group |first1=G. E. |last1=Murphy | year=1981 | journal=J. Algebra | volume=69 | pages=287–297 |doi=10.1016/0021-8693(81)90205-2 |issue=2 |doi-access=free

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