Capelli's Identity - Relations With Representation Theory - Case m = 1 and Representation Sk Cn

Case m = 1 and Representation Sk Cn

It is especially instructive to consider the special case m = 1; in this case we have x_i1, which is abbreviated as x_i:

In particular, for the polynomials of the first degree it is seen that:

Hence the action of restricted to the space of first-order polynomials is exactly the same as the action of matrix units on vectors in . So, from the representation theory point of view, the subspace of polynomials of first degree is a subrepresentation of the Lie algebra, which we identified with the standard representation in . Going further, it is seen that the differential operators preserve the degree of the polynomials, and hence the polynomials of each fixed degree form a subrepresentation of the Lie algebra . One can see further that the space of homogeneous polynomials of degree k can be identified with the symmetric tensor power of the standard representation .

One can also easily identify the highest weight structure of these representations. The monomial is a highest weight vector, indeed: for i < j. Its highest weight equals to (k, 0, ... ,0), indeed: .

Such representation is sometimes called bosonic representation of . Similar formulas define the so-called fermionic representation, here are anti-commuting variables. Again polynomials of k-th degree form an irreducible subrepresentation which is isomorphic to i.e. anti-symmetric tensor power of . Highest weight of such representation is (0, ..., 0, 1, 0, ..., 0). These representations for k = 1, ..., n are fundamental representations of .