The short version:

Can the theory of weights for SL(n,C) be explained concretely in terms of raising and lowering operators on spaces of polynomials?

A deleted question asked how to prove SL(3,C) acts irreducibly on the space of homogenous polynomials of fixed degree. I think it was deleted as a homework question, but it has never been a homework question at my university (at least not in the past decade), and so I thought it might be reasonable to try and work out these ideas in a setting that makes sense to me:

Let K be a field of characteristic 0 and let SL(n,K) be the group of n×n matrices with entries from K and determinant 1. Let V be the K-vector space with basis the monomials x^{i} of total degree w, where **i** = (i_{1},i_{2},...,i_{n}) is a multi-index with n entries, all non-negative integers whose sum is w. V has dimension w+1. SL(n,K) operates on V in the one true manner, by substitution of variables. In particular, for j = 2,3,...,n, the group SL(n,K) contains the elementary substitutions E_{j} which takes x_{j} to x_{1}+x_{j} and takes x_{k} to x_{k} when j≠k.

Define the lowering operator L_{j} as E_{j}−1, so it takes a polynomial f to the difference between E_{j}(f) and f itself. The L are called lowering operators because they lower the degree of f when considered as a polynomial in x_{j}, though they hold the total degree constant. This can be checked by applying it to monomials. For any individual monomial, factor it as x_{j}^{a}⋅x^{i} where i_{j}=0, then L_{j}(x_{j}^{a}⋅x^{i}) = (x_{1}+x_{j})^{a}.x^{i} − x_{j}^{a}⋅x^{i} = ( (x_{1}+x_{j})^{a}−x_{j}^{a} )⋅x^{i}. The leading term, when considered as a polynomial in x_{j} is then ax_{1}x_{j}^{a−1}x^{i}, so it has degree a−1 in x_{j}.

If W is a submodule of V containing some nonzero polynomial f, then it must have nonzero degree in one of the variables x_{j}. If any of these j are greater than 1, then apply L_{j} to transfer the degree from j to 1. Repeat until one is left with a⋅x_{1}^{w}. Since each L_{j} takes W to W and W is closed under division by nonzero scalars, we must have that any non-empty SL(n,K) invariant subspace of V contains x_{1}^{w}. In fact, we only use that it is closed under the maximal unipotent subgroup generated by the E_{j}.

For instance, if n=3, then we can more simply write x_{1}=x, x_{2}=y, and x_{3}=z. If w=4, and W contains xyyz+2xyzz, then apply L_{3} = L_{z} to get xyy(x+z)+2xy(x+z)(x+z) − xyyz+2xyzz = xyyx + 2xyxx + 4xyxz = 2xxxy + xxyy + 4xxyz. The total degree remained the same, but shifted from z to x, at the cost of increasing the leading coefficient. Applying L_{z} again, one gets the first two terms vanish, and one is left with 4xxy(x+z) − 4xxyz = 4xxxy. Applying L_{y} one gets the pure 4xxx(x+y)−4xxxy = 4xxxx.

Notice that there were multiple paths we could have taken to move from f to x^{w}. I think these paths are basically what are called "weights", and the lowering operators are called (simple, positive) "roots".

Now we must recover all other monomials from this monomial, and so we define the raising operators. We begin with the elementary F_{j} in SL(n,K) defined by taking x_{1} to x_{1}+x_{j} and x_{k} to x_{k} for k≠j. The raising operator R_{j} is then F_{j}−1 which takes a polynomial f to the difference between F_{j}(f) and f itself. R_{j} sacrifices a degree in x_{1} to provide a degree in x_{j}. Applying each R_{j} operator i_{j} times to x_{1}^{w} we obtain a polynomial with nonzero x^{i} term, but with a great many other terms that seem difficult to control.

How does one use the raising operator to recover the other monomials?

For instance, to show xxyz is in W, I imagine that we should apply R_{z} to xxxx to get (x+z)(x+z)(x+z)(x+z) − xxxx = 4xxxz + 6xxzz + 4xzzz + zzzz. But then, how does one isolate 4xxxz? Here is my current method, which seems overly complex compared to lowering: Applying L_{z} 3 times and subtract 60xxxx (already known to be in W) takes zzzz to a multiple of xxxz, but it does so by zero-ing out the other terms. Applying R_{y} to xxxz gives (x+y)(x+y)(x+y)z−xxxz = 3xxyz + 3xyyz + yyyz. Now apply L_{y} twice to zero out the first two terms and take yyyz to 12xxxz + 6xxyz. Since xxxz is known to be in W, we can remove it and rescale to get xxyz in W.

This random sort of zig-zag makes it hard to keep track of which "roots" we've been applying (that is, which raising and lowering operators). In the Lie algebra case, I thought things were a bit cleaner. At any rate, successfully applying these operators should have the side-effect of detecting in a very concrete way the "highest weight", if I understand correctly. However, I cannot yet check, since I cannot yet successfully keep track of how I am applying these operators.

Is there a clear relationship between the paths one takes using these operators and the theory of weights? For instance, is x

_{1}^{w}a highest weight vector?

It seems to me that SL(n,K) should have other irreducible representations other than just these polynomial representations, just because the paths defined by the operators are so symmetric. I think for SL(2,K) this is basically all there is, perhaps allowing for field automorphisms to be applied first.

Can all of the finite dimensional irreducible representations of SL(n,K) be explained in terms of actions on polynomials by substitutions and field automorphisms?

Assuming that this stuff makes sense, it would be handy to try it out on a different weight lattice.

Is there a version of this sort of description for any of the other classical groups or for G

_{2}?

It would be nice if say the symplectic group operated by substitutions on some subspace of homogenous polynomials, possibly restricted in some way, and that one could then find the raising and lowering operators. Any group where the action is natural and is likely to work well in the setup of Dickson's Linear Groups is fine, but I suspect the symplectic group might be the one with the least complications.

Dirac operators?"Outlined their answers"? OK, I have no idea what you mean or want and I am not sure you know it, either. I have answered three specific questions you had (last 3 of the 5 highlighted sentences), and any good book on semisimple Lie groups/algebras would explain what weights are and how raising and lowering operators, which are specific elements of the Lie algebra, are used to describe and classify irreducible finite-dimensional representations (the first 2 highlighted sentences). If you can compress your thoughts into a single precise question, repost it. Good luck! – Victor Protsak May 14 '10 at 2:04