[Notations are as in Macdonald's Symmetric Functions and Hall Polynomials]
The space of symmetric functions $\Lambda_{\mathbb{Q}}$ has a bilinear form defined by $ (p_\lambda, p_\mu)= z_\lambda \delta_{\lambda, \mu}$ for any two partitions $\lambda, \mu$ where for $\lambda = (1^{m_1} 2^{m_2}...)$ , $z_\lambda= \prod i^{m_i} m_i!$.
This definition can be motivated as follows: $\Lambda_n = \mathbb{C}[x_1,...,x_n]^{S_n}$ is the ring of characters of $\mathfrak{gl}_n(\mathbb{C})$. So $\Lambda_n$ is isomorphic to the ring of representations $R$ of $\mathfrak{gl}_n(\mathbb{C})$. On $R$ we have a natural bilinear form defined by $(V,W)=\dim Hom (V,W)$. By Schur's lemma, for two irreducible representations $V$ and $W$ we have $(V,W)=\delta_{V,W}$. Now, translating into characters this becomes $(s_\lambda,s_\mu)=\delta_{\lambda,\mu}$ for any two partitions $\lambda$, $\mu$ of length $\leq n$. This motivates us to define $(s_\lambda,s_\mu)=\delta_{\lambda,\mu}$ for any two partitions $\lambda, \mu$ (of any length) on the space $\Lambda$. Since $\{s_\lambda: \lambda \in Par\}$ is a $\mathbb{Z}$ basis for $\Lambda$, this defines a form on $\Lambda$. Now we can extend this form further to $\Lambda_{\mathbb{Q}}$. Then one can show $ (p_\lambda, p_\mu)= z_\lambda \delta_{\lambda, \mu}$ for any two partitions $\lambda, \mu$.
Now we extend the ring $\Lambda_{\mathbb{Q}}$ to $\Lambda_{\mathbb{Q}(q,t)}$. Here Macdonald defines a $q,t-$ analog of the form by defining $ (p_\lambda, p_\mu)= z_\lambda(q,t) \delta_{\lambda, \mu}$ for any two partitions $\lambda, \mu$; where now $z_\lambda(q,t)=z_\lambda \prod \dfrac{1-q^{\lambda_i} }{1-t^{\lambda_i}}$. This form reduces to the earlier one when $q=t$. When $q=0$ it reduces to the Hall-Littlewood form.
The change of basis matrix between the Schur functions and the monomial symmetric functions is triangular with $1$s on the diagonal. The Schur functions are characterized by this property and the property that they are orthogonal with respect to the form on $\Lambda_{\mathbb{Q}}$. The Macdonald symmetric functions are defined to be those with the property of uni-triangularity with respect to monomial symmetric function basis, and orthogonality with respect to the $q,t-$ form on $\Lambda_{\mathbb{Q}(q,t)}$. The existence need to be shown, and that is dealt with in Macdonald's book. The Macdonald polynomials specialize to various other known symmetric functions, partly because the form specializes to classical forms.
My question:
- Can we arrive at the definition of the $q,t-$ form naturally?
- If we instead take $z_\lambda(q,t)$ to be some other rational function in $q$ and $t$, which has some specialization properties (for e.g. at $q=t$ it should be $1$), and then try to find orthogonal basis with unitriangularity properties with respect to monomial symmetric function basis, will we arrive at some other 'interesting' symmetric functions? Or is the particular form the only form to study when seen from some point of view?
