The connection between symmetric functions and representation theory is well-known.


Now consider the subspace of symmetric functions that are shift-invariant,
that is, functions satisfying $f(x+t+y+t,z+t,\dots,)=f(x,y,z,\dots)$ for all $t$.


A simple example of such a function is the square of the Vandermonde determinant.


Such functions appear naturally in physics, but my question is:
*do shift-invariant symmetric functions appear (naturally) in representation theory?*



Shift-invariant symmetric functions in representation theory?
The shift invariant symmetric functions are naturally identified with the characters of the Lie algebra $\mathfrak{pgl}(n)$, the quotient of matrices by scalar matrices. You could rightly point out that I have an isomorphism of Lie algebras $\mathfrak{pgl}(n)\cong \mathfrak{sl}(n)$ (the latter is trace 0 matrices), but if you think about the latter, you'll write down the wrong formulas. For any representation of $\mathfrak{pgl}(n)$, the trace of the action of a matrix $A$ is a shift invariant symmetric function of its eigenvalues (since adding a constant to all the eigenvalues gives the same element of $\mathfrak{pgl}(n)$). Any shift invariant symmetric function can be written as a sum of these characters, though not uniquely, since the vector representation has trivial character (which is a lesson in the difference between Lie algebras and groups!).


**EDIT**: To understand the relationship to special functions like Schur functions: one way to think about this is in terms of normalized variables $\tilde{x}\_i=x\_i-(x\_1+\dots+x\_n)/n$. These are obviously shift invariant, and the shift invariant symmetric functions are just polynomials in the elementary symmetric functions of degree $\geq 2$ applied to this alphabet: $e\_2(\tilde{\mathbf{x}}),e\_3(\tilde{\mathbf{x}}), \dots$ (of course, $e\_1(\tilde{\mathbf{x}})=0$). The characters of simple $\mathfrak{pgl}(n)$ representations are just the Schur functions applied to $\tilde{\mathbf{x}}$.


