John Baez
Schur notes

This material from Schur functors I may still be useful.

The first thing that should be understood from the beginning is that a general Schur functor FF is nonlinear: the action on hom-sets

hom(V,W)hom(F(V),F(W))hom(V, W) \to hom(F(V), F(W))

is not assumed to respect the linear structure. In fact, linear Schur functors are rather uninteresting: because every finite-dimensional space is a finite direct sum of copies of the 11-dimensional space \mathbb{C}, and because linear functors preserve finite direct sums (that is, biproducts), it turns out that every linear Schur functor FF is representable as hom(X,)hom(X, -) where X=F()X = F(\mathbb{C}). So, the category of linear Schur functors is equivalent to FinVect\Fin\Vect.


John Baez: This stuff should get worked into the discussion near the end of how Schur functors are like polynomials…

Modules over a bimonoid

Next we exploit the fact that, just like any group algebra, k[S n]k[S_n] is a bialgebra — or in fancier language, a bimonoid in the symmetric monoidal category FinVect k\Fin\Vect_k. Since i:FinVect kCi : \Fin\Vect_k \to C is a symmetric monoidal functor, this means that ii carries k[S n]k[S_n] to a bimonoid in CC. As noted above, we call this bimonoid by the same name, k[S n]k[S_n].

The category of modules over a bimonoid is a monoidal category. More explicitly, in the case of the bimonoid k[S n]k[S_n] in CC with comultiplication

δ:k[S n]k[S n]k[S n],\delta: k[S_n] \to k[S_n] \otimes k[S_n] \,,

the tensor product VWV \otimes W of two k[S n]k[S_n]-modules in CC carries a module structure where the action is defined by

k[S n]VWδ1 VWk[S n]k[S n]VW1σ1k[S n]Vk[S n]Wα Vα WVWk[S_n] \otimes V \otimes W \stackrel{\delta \otimes 1_{V \otimes W}}{\to} k[S_n] \otimes k[S_n] \otimes V \otimes W \stackrel{1 \otimes \sigma \otimes 1}{\to} k[S_n] \otimes V \otimes k[S_n] \otimes W \stackrel{\alpha_V \otimes \alpha_W}{\to} V \otimes W

where σ\sigma is a symmetry isomorphism and α V\alpha_V, α W\alpha_W are the actions on VV and WW.

Now we consider a particular case of tensor product representations. If XX is an object of CC, the symmetric group S nS_n has a representation on X nX^{\otimes n}. (Indeed, for each σS n\sigma \in S_n, there is a corresponding symmetry isomorphism X nX nX^{\otimes n} \to X^{\otimes n}. From this one may construct an action

k[S n]X n σS nIX nX nk[S_n] \otimes X^{\otimes n} \cong \bigoplus_{\sigma \in S_n} I \otimes X^{\otimes n} \to X^{\otimes n}

which is the required representation.) So, if V νV_\nu is a Young tableau representation of k[S n]k[S_n] in CC, we obtain a tensor product representation

V νX nV_\nu \otimes X^{\otimes n}

of k[S n]k[S_n] in CC.

Consider next the averaging operator e=1n! σS nσe = \frac1{n!} \sum_{\sigma \in S_n} \sigma:

e:V νX nV νX ne: V_\nu \otimes X^{\otimes n} \to V_\nu \otimes X^{\otimes n}

This operator makes sense since kk has characteristic zero, and crucially, this operator is idempotent (because e=σee = \sigma e for all σS n\sigma \in S_n). Because we assume idempotents split in CC, we have a (split) coequalizer

V νX n1eV νX nV ν S nX nV_\nu \otimes X^{\otimes n} \stackrel{\overset{e}{\to}}{\underset{1}{\to}} V_\nu \otimes X^{\otimes n} \to V_\nu \otimes_{S_n} X^{\otimes n}

This coequalizer is indeed the object of S nS_n-coinvariants of V νX nV_\nu \otimes X^{\otimes n}, i.e., the joint coequalizer of the diagram consisting of all arrows

V νX nσV νX nV_\nu \otimes X^{\otimes n} \stackrel{\sigma \cdot-}{\to} V_\nu \otimes X^{\otimes n}

ranging over σS n\sigma \in S_n (it is the joint coequalizer because e=σee = \sigma e for all σ\sigma).

We may now define the Schur functor S νS_\nu on CC attached to a Young tableau ν\nu.

Created on July 31, 2013 at 16:39:55. See the history of this page for a list of all contributions to it.