Jucys–Murphy element

They generate a commutative subalgebra of $\backslash mathbb\{C\}\; [\; S\_n]$. Moreover, X_n commutes with all elements of $\backslash mathbb\{C\}\; [S\_\{n-1\}]$.

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:

:$X\_k\; v\_U\; =c\_k(U)\; v\_U,\; ~~~\; k=1,\backslash dots,n,$

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 $Z(\backslash mathbb\{C\}\; [S\_n])$ of the group algebra $\backslash mathbb\{C\}\; [S\_n]$ of the symmetric group is generated by the symmetric polynomials 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 $\backslash mathbb\{C\}\; [S\_n]$ holds true:

:$(t+X\_1)\; (t+X\_2)\; \backslash cdots\; (t+X\_n)=\; \backslash sum\_\{\backslash sigma\; \backslash in\; S\_n\}\; \backslash sigma\; t^\{\backslash text\{number\; of\; cycles\; of\; \}\backslash sigma\}.$

Theorem (Okounkov–Vershik): The subalgebra of $\backslash mathbb\{C\}\; [S\_n]$ generated by the centers

: $Z(\backslash mathbb\{C\}\; [\; S\_1]),\; Z(\backslash mathbb\{C\}\; [\; S\_2]),\; \backslash ldots,\; Z(\backslash mathbb\{C\}\; [\; S\_\{n-1\}]),\; Z(\backslash mathbb\{C\}\; [S\_n])$

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

• Representation theory of the symmetric group
• Young symmetrizer

• {{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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Category:Permutation groups

Category:Representation theory

Category:Symmetry

Category:Representation theory of finite groups

Category:Symmetric functions