A Review On FI-modules And Stability For Representations Of Symmetric Groups

This paper is a review on the stability for representations of symmetric groups and FI-modules.We introduce the notion and applications of FI-modules,FI#-modules and graded FI-modules.By defining the weight,the slope and the stable degree on these modules,someone can build up a theory which uses FI-modules to describe the stability for the representations of symmetric groups.This theory converts the representation stability(in the sense of Church-Farb)for a sequence of Sn-representations to a finite generation property for a single FI-module,which makes it easier for us to prove the representation stability.We reviewed how to use this theory to prove the representation stability of Schur functors and give a new proof of the Murnaghan theorem.We also reviewed the application of this theory to the cohomology of the configuration space of n distinct ordered points on an arbitrary(connected,oriented)manifold,the diagonal coinvariant algebra on r sets of n variables,the space of polynomials on rank varieties of n×n matrices.The symmetric group Sn acts on each of these vector spaces.In most cases almost nothing is known about the characters of these representations,or even their dimensions.This theorem shows that in each fixed degree the character is given,for n large enough,by a polynomial in the cycle-counting functions that is independent of n.In particular,the dimension is eventually a polynomial in n.It also gives a more precise description of this representation and replace the "sufficiently large n" with an explicit stable range.