From: jcbhrb@nic.cerf.net (Jacob Hirbawi)
Newsgroups: sci.math
Subject: su(m) and the symmetric group
Date: 21 Feb 1995 01:27:47 GMT
There's a well known correspondance between the symmetric group Sym(n)
and the Lie algebra su(m); and then there's this one! It's a curious
way of defining elements in the group algebra of Sym(n) (over Q or C)
that satissfy the commutation relations of su(m). For example in the group
algebra of Sym(3) define the three elements : h1,e12,e21 :
3*h1 = +2*(1,2)-(1,2,3)+(1,3,2)-2*(1,3)
3*e12 = +(2,3)-(1,2)+(1,2,3)-(1,3,2)
3*e21 = +(2,3)-(1,2,3)+(1,3,2)-(1,3)
and the commutator [x,y]= x*y - y*x then you have :
[h1 ,e21] = 2 e12, [h1 ,e12] = -2 e21, [e21,e12] = h1
which are the commutation relations of su(2)!
The process of finding these elements is in fact not too hard. It also
works for any finite group and not just the symmetric group (the choice
of the symmetric group in the subject line is for dramatic purposes only!) :
you start with an explicit irreducible matrix representation of the group of
dimension m, you then form the m^2 "projection operators" :
P_{ij} = Sum M_{ij}(g) g^-1 i,j=1..m
where the sum is over all the elements g of the group; M_{ij}(g) is the
ij-the entry in the matrix that represents g. The P_{ij}'s turn out to
satissfy the commutations of u(m); taking out the center gives su(m)!
For those interested, there's a paper by A. Gamba in Journal of Mathematical
Physics, vol 10, no. 5, May 1969, p 872-874 that talks about this. Also for
those really interested, there's an explicit base for su(3) in the group
algebra of Sym(4) at the end of this.
Jacob Hirbawi
8*h1 = -(3,4)+(2,4)+2*(1,2)-(1,2)(3,4)-(1,2,3)+(1,2,3,4)+(1,2,4,3)-2*(1,2,4)
+(1,3,2)-(1,3,4,2)-2*(1,3)+2*(1,3,4)+(1,3)(2,4)-(1,3,2,4)-(1,4,2)+(1,4,3)
8*h2 = -(3,4)+2*(2,3)-(2,4)+2*(1,2)-(1,2)(3,4)-3*(1,2,3)+(1,2,3,4)+(1,2,4,3)
-3*(1,3,2)+(1,3,4,2)+2*(1,3)-(1,3)(2,4)+(1,3,2,4)-2*(1,4,3,2)+3*(1,4,2)
+3*(1,4,3)-4*(1,4)-2*(1,4,2,3)+2*(1,4)(2,3)
8*e12 = +(2,3)-(2,4,3)-(1,2)+(1,2)(3,4)+(1,2,3)-(1,2,3,4)-(1,2,4,3)+(1,2,4)
-(1,3,2)+(1,3,2,4)+(1,4,3,2)-(1,4)(2,3)
8*e13 = +(2,3,4)-(2,4)+(1,2)-(1,2)(3,4)-(1,2,3)+(1,2,3,4)+(1,2,4,3)-(1,2,4)
-(1,3,4,2)+(1,3)(2,4)+(1,4,2)-(1,4,2,3)
8*e21 = +(2,3)-(2,3,4)-(1,2,3)+(1,2,3,4)+(1,3,2)-(1,3,4,2)-(1,3)+(1,3,4)
+(1,3)(2,4)-(1,3,2,4)+(1,4,2,3)-(1,4)(2,3)
8*e23 = +(3,4)-(2,4,3)-(1,2)(3,4)+(1,2,4,3)+(1,3,2)-(1,3,4,2)-(1,3)+(1,3,4)
+(1,3)(2,4)-(1,3,2,4)+(1,4,3,2)-(1,4,3)
8*e31 = +(2,4,3)-(2,4)-(1,2,4,3)+(1,2,4)+(1,3)(2,4)-(1,3,2,4)+(1,4,3,2)
-(1,4,2)-(1,4,3)+(1,4)+(1,4,2,3)-(1,4)(2,3)
8*e32 = +(3,4)-(2,3,4)-(1,2)(3,4)+(1,2,3,4)+(1,3,4,2)-(1,3,4)-(1,4,3,2)
+(1,4,2)+(1,4,3)-(1,4)-(1,4,2,3)+(1,4)(2,3)
==============================================================================
From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: su(m) and the symmetric group
Date: 21 Feb 1995 17:54:42 GMT
In article <3ibfij$mo0@news.cerf.net>,
Jacob Hirbawi wrote:
>There's a well known correspondance between the symmetric group Sym(n)
>and the Lie algebra su(m); and then there's this one! It's a curious
>way of defining elements in the group algebra of Sym(n) (over Q or C)
>that satissfy the commutation relations of su(m). For example in the group
>algebra of Sym(3) define the three elements : h1,e12,e21 :
>
> 3*h1 = +2*(1,2)-(1,2,3)+(1,3,2)-2*(1,3)
> 3*e12 = +(2,3)-(1,2)+(1,2,3)-(1,3,2)
> 3*e21 = +(2,3)-(1,2,3)+(1,3,2)-(1,3)
>
>and the commutator [x,y]= x*y - y*x then you have :
>
> [h1 ,e21] = 2 e12, [h1 ,e12] = -2 e21, [e21,e12] = h1
>
>which are the commutation relations of su(2)!
I don't quite see what the point is. If you look at the regular representation
of Sym(n), you get a direct sum of matrix rings. The three elements you've
quoted all map to zero in the linear representations, so that you're
listing elements in the matrix ring M_2(C); of course you can find
elements in this ring corresponding to the standard basis elements of the
Lie algebra in here; the poster has done so in some way.
More generally, if G is any group consider its regular representation
(or indeed any representation over C). Then the image is a direct sum of
matrix rings, so you can find elements of C[G] mapping to any set of
elements in any of these matrix rings you'd like, such as the basis for
sl(k,C) where k is the degree of the irreducible constituent in question.
(The symmetric groups are nicest because all the idempotents in C[G]
actually have rational coefficients.)
I'm not sure what the other connection between Sym(n) and su(m) is
supposed to be; apart from a study of Weyl groups I'd assume the
poster means the method of finding representations of each as the
commutator of the other. I never thought I'd get to do this but I'll
quote from my own undergrad thesis, which opens:
"Once the theory of the representations of the symmetric groups had been
worked out, it was noticed by Issai Schur [his 1901 dissertation] that
there was a close connection between these representations and those of
the general linear group GL(n,C)...Later, Hermann Weyl showed that the
representations obtained in this way were essentially the only ones..."
The basic idea is to observe that GL(n,C) acts naturally on the vector
space C^n and then by diagonal action on the d-fold tensor
product (C^n x C^n x ... x C^n). On the other hand, Sym(d) also
acts on this tensor product (by permuting the basis vectors) and
the two actions commute. Thus inside M_(nd) (C) we find two semisimple
algebras which are each other's commutators. For example, the action
of GL_2 on C^4 thus splits into a 3-dimensional representation (on the
span of e1xe1, e2xe2, and (e1xe2+e2xe1) ) and a 1-dimensional rep -- namely
the determinant -- on the span of e1xe2-e2xe1; these subspaces are
respectively the +1 and -1 representation spaces of S_2.
I believe Boerner's representation theory book has all this in it.
The statement that all ("rational") representations of GL(n,C) and Sym(d)
arise in this way is mirrored by a statement in the representation of finite
groups: if rho is any faithful representation, then the irreducible
components of the tensor product rho^N include all irreducible
representations for N sufficiently large.
(Apologies for having had to switch notation mid-post).
dave
==============================================================================
Date: Mon, 1 May 95 09:19:17 CDT
From: rusin (Dave Rusin)
To: gittelma@math.berkeley.edu
Subject: Re: f.d. reps of GL(n,R)
Newsgroups: sci.math.research
In article <3nrequ$58d@agate.berkeley.edu> you write:
>Is there a systematic description of all the finite dimensional
>continuous representations of GL(n,R) on real vector spaces?
>If so, where is this described?
Well, the representations GL(n) --> GL(m) which are polynomial
functions of the coordinates in GL(n) are direct sums of irreducible
summands of the following: given A in GL(n), you already have A
acting on V = R^n. Then A acts on the k-fold tensor product
V^k by diagonal action. This gives a representation of GL(n) into
GL(nk). The image of this representation is precisely the
commutator of the action of the symmetric group S_k on V^k
which acts by permuting the tensor factors. Since this representation
of S_k is semisimple, so is the representation of GL(n);
indeed, if the rep. of S_k is Sum ( e_i rho_i) where the
rho_i are distinct and the e_i are the multiplicities, then
the rep. of GL(n) decomposes as Sum(f_i tau_i) where
e_i = dim(tau_i) and f_i = dim(rho_i).
Example: n=k=2: the representation of GL(2) into GL(4) is given by
16 quadratic polynomials in 4 variables, but if a new basis is
chosen so that the representation of S_2 is a diagonal matrix
diag(1,1,1,-1), then the representation of GL(2) becomes a direct
sum of a 3x3 block and a 1x1 block; the last is the determinant.
Of course you will not in this way produce any representations such
as the map GL(1) --> GL(2) given by x --> matrix( (1,log|x|), (0,1) )
I learned this material in Boerner's book on group representations.
dave
==============================================================================
From: cfeaux@Calvados.MI.Uni-Koeln.DE (Tobias Feaux de Lacroix)
Newsgroups: sci.math.research
Subject: Q: Embeddings of SL_2 in SL_n
Date: 6 Jul 1995 12:50:38 GMT
>Can somebody tell me which embeddings, i.e.
>injective (locally analytic) homomorphisms
>of SL_2 in SL_n can occur?
...
>I'm interested in the case of a finite extension of Q_p,
>but results about C or R are also welcome.
Using an idea of Dave Rusin I was able to gain a family f_n of 'embeddings' of
SL_2 in SL_n (see below):
On Thu, 22 Jun 1995, Dave Rusin wrote:
Well, there's the general representation theory of GL_n which constructs
all the representations of GL_n into GL_m whose coordinates are
polynomials in the original entries (for example, the determinant
representation GL_n --> GL_1). These restrict to representations of
SL_n and, apart from the powers of the determinant map, must be
injective or nearly so since SL_n is close to simple.
The representations arise from taking the action of GL_n on a vector
space V and using that to give a (diagonal) action on V\tensor V or
in general on the k-fold tensor product of V with itself. This
action clearly commutes with the action of the symmetric group Sym(2)
(resp. Sym(k).) Thus if a basis of V\tensor V is chosen in which
the action of Sym(2) is block-diagonal, the action of GL_n will also
decompose into pieces, each of which turns out to be irreducible.
Try this for m=2: GL_n acts on V = F^2 (F is any field), but we let
it act diagonally on V\tensor V = F^4. If we take a basis {e1, e2}
of V,
then e1\tensor e2 - e2\tensor e1 spans the 1-dimensional -1 eigenspace
for Sym(2), while e1\tensor e1, e2\tensor e2, and e1\tensor
e2+e2\tensor e1
span the 3-dimensional +1 eigenspace for Sym(2). The action of GL_2
on the -1 eigenspace is simply multiplication by the determinant.
The action of GL_2 on the +1 eigenspace is given by a 3x3 matrix whose
entries are quadratic polynomials in the four entries of the original
matrix in GL_2.
> dave
Consider the representation of SL_2 on the homogenious polynomials of
degree n-1 of two variables U, V:
| a b | in SL_2 maps U |--> U a + V c
| c d | and V |--> U b + V d
Choose any basis; one gets a homomorphism which components are homogeneous
polynomials of degree n-1 in a,b,c,d. It is injective iff n even.
Combinig these and composing them with automorphisms of the target gp one
gets a large family of embeddings, charakterised by the fact that every
BOREL subgp has a eigen vector.
Because the field isn't supposed to be algebraically closed, I don't know
if there can be finite dimensional representations of SL_2 WITHOUT such
highest weight vectors.
Tobias