Alternative definition of complete homogeneous symmetric functions

Question

I found this definition of symmetric functions: $g_n=\sum\limits_{i_1\leq i_2\leq ... \leq i_n} x_{i_1}x_{i_2}...x_{i_n}$ where for each integer $j$ at most $t$ of the numbers $i_1,i_2,...$ are equal to $j$. Here $t$ is fixed. So that means for $t=1$ one obtains the elementary symmetric functions and for $t\geq n$ the homogeneous symmetric functions. So of course $g_n$ form a basis of the space of symmetric functions. But how about $g_{\lambda}$ with a partition $\lambda$? I feel like it should be enough to form a basis but i dont know how to proove it.

Answer 1

I it not clear what you mean by "of course the $g_n$ form a basis of the space of symmetric functions" (and by the way they are usually written $h_n$). They don't form a basis of the vector space (over the base field) of symmetric functions, but they do form an algebraically independent generating set of the (associative commutative) algebra of symmtric functions; this means one needs polynomial expressions (over the base field) in the $h_n$ to express and arbitrary symmetric function, and such expressions are unique. This means precisely that the monomials in the $h_n$ form a vector space basis, and these monomials are traditinally written $h_\lambda$, which by definition is $h_{\lambda_1}h_{\lambda_2}h_{\lambda_3}h_{\lambda_4}\ldots$.