Introduction
Let's fix minmathbbNminmathbbN. For each n, the unitary group mathbfU(m)mathbfU(m) is represented in the space of tensors of rank nn over mathbbCmmathbbCm
Vn,m=bigotimesnk=1mathbbCmVn,m=bigotimesnk=1mathbbCm and the symmetric group SnSn acts on Vn,mVn,m by permutation of factors. Now the space Vn,mVn,m breaks into the direct sum of subspaces Vn,m(lambda)Vn,m(lambda) which are primary with respect to each of these actions and irreducible with respect to the joint action of SntimesmathbfU(m)SntimesmathbfU(m)
Vn,m=bigopluslambdainmathbbY(n,m)Vn,m(lambda)Vn,m=bigopluslambdainmathbbY(n,m)Vn,m(lambda)
Where lambdalambda ranges over all Young diagrams of size nn with at most mm rows (So lambdainmathbbY(n,m)lambdainmathbbY(n,m) means lambda=(lambda1,dots,lambdak)lambda=(lambda1,dots,lambdak) is a partition of nn with klemklem). The tensors from Vn,m(lambda)Vn,m(lambda) are said to have symmetry type lambdalambda. We define the relative dimensions
dn,m(lambda)=fracdimVn,m(lambda)dimVn,mdn,m(lambda)=fracdimVn,m(lambda)dimVn,m
which tell us how tensors are distributed into symmetry types.
Motivation and Question
I was reading Kerov's "Asymptotic representation theory of the symmetric group and it's aplications in analysis" and was trying to provide proofs for some of the results stated there. (He does give references, which I can't reach at the moment.)
The following two theorems are due to Kerov
Theorem 1 If for each lambdalambda we associate x=(x1,dots,xm)x=(x1,dots,xm) with xk=fraclambda(n)k−n/msqrtnxk=fraclambda(n)k−n/msqrtn. The joint distribution of xkxk's as ntoinftyntoinfty with respect to the measure dn,mdn,m on mathbbY(n,m)mathbbY(n,m) weakly converges to an absolutely continuous measure on the cone Cm=x:x1geqx2geqcdotsgeqxm;;;sumxk=0Cm=x:x1geqx2geqcdotsgeqxm;;;sumxk=0
with density
phim(x)=cprodi<j(xi−xj)2e−m/2sumx2kphim(x)=cprodi<j(xi−xj)2e−m/2sumx2k where
c=fracm(m−1)m/21!2!cdots(m−1)!left(fracm2piright)(m−1)/2c=fracm(m−1)m/21!2!cdots(m−1)!left(fracm2piright)(m−1)/2
.
Theorem 2 Let lambda(n)inmathbbY(n,m)lambda(n)inmathbbY(n,m) be the Young diagram for which the tensors of type lambda(n)lambda(n) are most probable. Then limntoinftyfraclambda(n)k−n/msqrtn/m=zklimntoinftyfraclambda(n)k−n/msqrtn/m=zk
for each k=1,2,dots,mk=1,2,dots,m where z1,z2,dotszmz1,z2,dotszm are the roots of the Hermite polynomial Hm(z)Hm(z)
I can prove Theorem 2 assuming Theorem 1, but I don't see a nice argument for proving the first theorem itself. Can anyone provide a sketch of the proof, or some hint how to approach Theorem 1?
No comments:
Post a Comment