In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Jucys–Murphy elements in the
group algebra of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
, named after
Algimantas Adolfas Jucys
Algimantas Adolfas Jucys (14 November 1936 – 29 July 1997) was a Lithuanian theoretical physicist more prominent as a mathematician, a son of Lithuanian physicist Adolfas Jucys. Since 1967 Algis Jucys was researcher at the Institute of Physics ...
and G. E. Murphy, are defined as a sum of
transpositions by the formula:
:
They play an important role in the
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
.
Properties
They generate a commutative subalgebra of
. Moreover, ''X''
''n'' commutes with all elements of
.
The vectors constituting the basis of Young's "seminormal representation" are eigenvectors for the action of ''X''
''n''. For any
standard Young tableau In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and ...
''U'' we have:
:
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
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of the group algebra
of the symmetric group is generated by the
symmetric polynomial
In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has ...
s in the elements ''X
k''.
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
holds true:
:
Theorem (
Okounkov–
Vershik): The subalgebra of
generated by the centers
:
is exactly the subalgebra generated by the Jucys–Murphy elements ''X
k''.
See also
*
Representation theory of the symmetric group
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from s ...
*
Young symmetrizer In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space V^ obtained from the action of S_n on ...
References
*
*
*
*
*
{{DEFAULTSORT:Jucys-Murphy Element
Permutation groups
Representation theory
Symmetry
Representation theory of finite groups
Symmetric functions