Specht module: Difference between revisions

117 bytes added ,  3 months ago
→‎Definition: clarified things by explaining them in order
(Bluelinking 1 books for verifiability.) #IABot (v2.1alpha3)
(→‎Definition: clarified things by explaining them in order)
 
==Definition==
Fix a partition &lambda; of ''n''. This determines a [[Young diagram]] with ''n'' boxes. A [[Young tableau]] of shape &lambda; is a way of labelling the boxes of this Young diagram by distinct numbers <math>1, \dots, n</math>.
Fix a partition &lambda; of ''n''.
 
A '''tabloid''' is an equivalence class of labellings of the [[Young diagram]] of shape &lambda;, where two labellings are equivalent if one is obtained from the other by permuting the entries of each row.
A '''tabloid''' is an equivalence class of Young tableaux where two labellings are equivalent if one is obtained from the other by permuting the entries of each row. For each Young tableau ''T'' of shape &lambda; let <math>\{T\}</math> be the corresponding tabloid. The symmetric group on ''n'' points acts on the set of Young tableaux of shape &lambda;. Consequently, it acts on tabloids, and on the vector space ''V'' with the tabloids as basis.
Denote by <math>\{T\}</math> the equivalence class of a tableau <math>T</math>.
 
The symmetric group on ''n'' points acts on the set of tableaux of shape &lambda; (i.e., on the set of labellings of the Young diagram).
Consequently, it acts on tabloids, and on the module ''V'' with the tabloids as basis. ForGiven eacha [[Young tableau]] ''T'' of shape &lambda;, form the element let
:<math>E_T=\sum_{\sigma\in Q_T}\epsilon(\sigma)\{\sigma(T)\} \in V</math>
where ''Q''<sub>''T''</sub> is the subgroup of permutations, preserving (as sets) all columns of ''T'' and <math>\epsilon(\sigma)</math> is the sign of the permutation &sigma;. The Specht module of the partition &lambda; is the module generated by the elements ''E''<sub>''T''</sub> as ''T'' runs through all tableaux of shape &lambda;.
The Specht module of the partition &lambda; is the module generated by the elements ''E''<sub>''T''</sub> as ''T'' runs through all tableaux of shape &lambda;.
 
The Specht module has a basis of elements ''E''<sub>''T''</sub> for ''T'' a [[standard Young tableau]].