Jucys–Murphy element
Encyclopedia
In mathematics
, Jucys–Murphy elements in the group algebra
of the symmetric group
, named after Algimantas Adolfas Jucys
and G. E. Murphy, are defined by the formula:
They play an important role in the representation theory
of the symmetric group
.
They generate a commutative subalgebra of
. Moreover,
Xn commutes with all elements of
.
The vectors of the Young basis are eigenvectors for the action of Xn. For any standard Young tableau
U we have:
where ck(U) is content b − a of the cell (a, b) occupied by k in the standard Young tableau
U.
Theorem (Jucys): The center
of the group algebra
of the symmetric group
is generated by the symmetric polynomials in 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
holds true:
Theorem (Okounkov–Vershik): The subalgebra of
generated by the center
s
is exactly the subalgebra generated by the Jucys–Murphy elements Xk.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Jucys–Murphy elements in the group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
of the symmetric groupSymmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
, named after Algimantas Adolfas Jucys
Algimantas Adolfas Jucys
Algimantas Adolfas Jucys was a Lithuanian theoretical physicist more prominent as a mathematician, a son of Lithuanian physicist Adolfas Jucys...
and G. E. Murphy, are defined by the formula:
They play an important role in the representation theory
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 studiesmodules over these abstract algebraic structures...
of the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
.
They generate a commutative subalgebra of
. Moreover,Xn commutes with all elements of
.The vectors of the Young basis are eigenvectors for the action of Xn. For any standard Young tableau
Young tableau
In mathematics, a Young tableau is a combinatorial object useful in representation theory. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at...
U we have:
where ck(U) is content b − a of the cell (a, b) occupied by k in the standard Young tableau
Young tableau
In mathematics, a Young tableau is a combinatorial object useful in representation theory. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at...
U.
Theorem (Jucys): The center
Center (algebra)
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. It is often denoted Z, from German Zentrum, meaning "center". More specifically:...
of the group algebraGroup algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
of the symmetric groupSymmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
is generated by the symmetric polynomials in 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
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
holds true:Theorem (Okounkov–Vershik): The subalgebra of
generated by the centerCenter (algebra)
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. It is often denoted Z, from German Zentrum, meaning "center". More specifically:...
s
is exactly the subalgebra generated by the Jucys–Murphy elements Xk.