**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 .

Read more about this topic: Capelli's Identity, Relations With Representation Theory

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