Macdonald Polynomials

The combinatorial Macdonald polynomials $H_\mu(\mathbf{x};q,t),$ for $\mu$ partition of $n,$ are symmetric functions (polynomials in a denumerable set of variables) in variables $\mathbf{x}=x_1,x_2,\ldots$, with coefficients in the field of fraction $\mathbb{Q}(q,t).$ Their Schur Function expansions have coefficients that are polynomials in $\mathbb{N}[q,t],$ hence:

$H_\mu(\mathbf{x};q,t) = \sum_{\nu\vdash n} k_{\mu\nu}(q,t)\, s_\nu(\mathbf{x}),$

with $k_{\mu\nu}(q,t)\in\mathbb{N}[q,t]$. At $q=t=1$ they specialize to $s_1(\mathbf{x})^n$. For example,

$H_{{3}}(\mathbf{x};q,t)=s_{{3}}(\mathbf{x})+ \left( q+{q}^{2} \right) s_{{21}}(\mathbf{x})+{q}^{3}s_{{111}}(\mathbf{x})$
$H_{{21}}(\mathbf{x};q,t)=s_{{3}}(\mathbf{x})+ \left( t+q \right) s_{{21}}(\mathbf{x})+qt\,s_{{111}}(\mathbf{x})$
$H_{{111}}(\mathbf{x};q,t)=s_{{3}}(\mathbf{x})+ \left( t+{t}^{2} \right) s_{{21}}(\mathbf{x})+{t}^{3}s_{{111}}(\mathbf{x})$

Many interesting operators are linked to the study of these polynomials, in particular the operator $\nabla$.

The combinatorial Macdonald polynomial $H_\mu(\mathbf{x};q,t)$ occur as the bigraded Frobenius characteristic of the Garsia-Haiman module $M_\mu$. This means that the coefficient of $q^i\,t^j$ in $k_{\mu\nu}(q,t)$ is the multiplicity of the irreducible representation of $\mathbb{S}_n$ corresponding to the partition $\nu$, in the homogenous component of bidegree $(i,j)$ of $M_\mu$. The $\mathbb{S}_n$-module $M_\mu$ is a (bi-)homogeneous subvector space of the ring of polynomials in two sets of $n$ variables.

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