# Nonsymmetric Macdonald Polynomials


## Background

Since we will be restricting ourselves to $$GL_n$$, we consider the weight lattice $$X = \mathbb{Z}^n$$, with simple roots $$\alpha_i = e_i - e_{i+1}$$, where $$e_i$$ is the $$i$$-th unit vector. We identify the co-weight lattice $$X^{\vee}$$ with $$X$$ using the standard inner product on $$\mathbb{Z}^n$$ so that $$\alpha^{\vee}_i = \alpha_i$$. Therefore the dominant weights are $$\langle \lambda, \alpha_i^{\vee} \rangle \geq 0$$ for all $$i$$ and are partitions.

The affine weight lattice is $$\widehat{X} = X \oplus \mathbb{Z} \delta$$ where $$\delta$$ is the smallest positive imaginary root, or null root. The extra simple root is $$\alpha_0 = \delta - \theta$$ where $$\theta = e_1 - e_n$$ is the highest root of $$GL_n$$. The positive affine roots are $$\widehat{R}_+ = \{ e_i - e_j + k\delta \mid i \neq j, k > 0, \text{ and if } i > j, k > 0 \}$$. We denote $$x^{\delta} = q$$, so $$x^{\alpha_i} = x_i /x_{i+1}$$ and $$x^{\alpha_0} = q x_1 / x_n$$. Therefore the group ring $$\mathbb{Q}(t)\widehat{X} \subseteq \mathbb{Q}(q,t)X$$ by extending scalars.

The inner product we want is Cherednik's inner product on $$\mathbb{Q}(q,t)X$$ given by $\langle f, g \rangle_{q,t} = [x^0](f \bar{g} \Delta_1)$ where $$\bar{\cdot}$$ is the involution given by $$\bar{q} = q^{-1}, \bar{t} = t^{-1}, \bar{x}_i = x_i^{-1}$$, and $$\Delta_1 = \Delta / ([x^0](\Delta)$$ with $\Delta = \prod_{\alpha \in \widehat{R}_+} \dfrac{1 - x^{\alpha}}{1 - t x^{\alpha}} = \prod_{i<j} \prod_{k=0}^{\infty} \dfrac{(1 - q^k x_i / x_j) (1 - q^{k+1} x_j / x_i)}{(1 - t q^k x_i / x_j) (1 - t q^{k+1} x_j / x_i)}.$ Here $$[x^\alpha] f$$ denotes the coefficient of $$x^{\alpha}$$ in $$f \in \mathbb{Q}[[q,t]]$$. It is known that $$[x^{\lambda}] \Delta \in \mathbb{Q}(q,t)$$ and $$\overline{\Delta}_1 = \Delta_1$$. Hence $$\langle f, g \rangle_{q,t} \in \mathbb{Q}(q,t)$$ and $$\langle f, g \rangle_{q,t} = \overline{\langle g, f \rangle}_{q,t}$$.

The Bruhat order on $$X$$ is given by identification with $$\widehat{W} / W_0$$ where $$W_0 = S_n$$ is the Weyl group of $$GL_n$$ and $$\widehat{W} = W_0 \ltimes X$$ is the extended affine Weyl group, equipped with the usual Bruhat order. Explicitly for $$GL_n$$ we have $$\lambda > \sigma_{ij}(\lambda)$$ if $$i < j$$ and $$\sigma_{ij}$$ is the transposition $$(i \; j)$$. If $$\lambda_i < \lambda_j$$ and $$\lambda_j - \lambda_i > 1$$, then $$\sigma_{ij}(\lambda) > \lambda + e_i - e_j$$.

We define the nonsymmetric Macdonald polynomials $$E_{\mu}(x; q, t) \in \mathbb{Q}(q,t) X$$ for $$\mu \in X$$ are uniquely characterized by the conditions:

1. Triangularity: $$E_{\mu} \in x^{\mu} + \mathbb{Q}(q, t)\{ x^{\lambda} \mid \lambda < \mu \}.$$
2. Orthogonality: $$\langle E_{\lambda}, E_{\mu} \rangle_{q,t} = 0$$ for $$\lambda \neq \mu,$$

One last note, the notation used here might differ from that used elsewhere.

## Hecke Algebras

The affine Hecke algebra is the $$\mathbb{Q}(t)$$-algebra $$\mathcal{H} = \langle T_0, T_1, \ldots, T_{n-1} \rangle$$ which satisfy the braid relations $T_i T_{i+1} T_i = T_{i+1} T_i T_{i+1} \\ T_i T_j = T_j T_i \hspace{20pt} i - j \neq \pm 1,$ where indices are taken modulo $$n$$, and the quadratic relation $(T_i - t)(T_i + 1) = 0.$ The (unextended)​ affine Weyl group $$W_a = \langle s_0, s_1, \ldots s_{n-1} \rangle$$ which satisfiy the braid relations above and act naturally on $$\widehat{X}$$, as well as the extensions, by $s_i(\lambda) = \lambda - \langle \lambda, \alpha_i^{\vee} \rangle \alpha_i.$ Explicitly for $$i \neq 0$$, these are the usual transpositions, and $s_0 f(x_1, \ldots, x_n) = f(q x_n, x_2, \ldots, x_{n-1}, x_1/q).$

Cherednik's representation of $$\mathcal{H}$$ is given by the formula $T_i x^{\lambda} = t x^{s_i(\lambda)} + (t - 1) \dfrac{x^{\lambda} - x^{s_i(\lambda)} }{1 - x^{\alpha_i}}.$ If we now define operators $$Y^{\beta} = t^{-\langle \beta, \rho^{\vee} \rangle} T_{\tau(\beta)}$$ where $$\{T_w\}_{w \in W_a}$$ is the standard basis for $$\mathcal{H}$$ and $$\tau(\beta)$$ is the translation by $$\beta$$ and $$\rho^{\vee} = \sum_{\alpha \in R^+} \alpha^{\vee} / 2$$. The operators $$Y^{\beta}$$ commute, are unitary with respect to $$\langle \cdot, \cdot \rangle_{q,t},$$ and are lower triangular with respect to Bruhat order on the basis $$\{x^{\lambda}\}_{\lambda \in X}.$$

## Nonsymmetric Macdonald Polynomials

From Bernstein's relations in $$\mathcal{H}$$, the simultaneous eigenfunctions $$E_{\mu}(x; q, t)$$ statisfy the relations $E_{s_i(\mu)}(x; q, t) = \left( T_i + \dfrac{1 - t}{1 - q^{\langle \mu, \alpha_i^{\vee} \rangle} t^{\langle w_{\mu}(\rho), \alpha_i^{\vee} \rangle}} \right) E_{\mu}(x; q, t)$ for $$i \neq 0$$, $$\mu_i > \mu_{i+1}$$ and where $$w_{\mu} \in W_0$$ is the maximal length permutation such that $$w_{\mu}^{-1}(\mu)$$ is dominant. Next we need the following automorphisms $\pi(\lambda_1, \ldots \lambda_n) = (\lambda_n+1, \lambda_1, \ldots, \lambda_{n-1}) \\ \Psi f(x_1, \ldots, x_n) = x_1 f(x_2, \ldots x_n, q^{-1} x_1).$ We can show that $E_{\pi(\mu)}(x; q, t) = q^{\mu_n} \Psi E_{\mu}(x; q, t),$ and these two relations are known as the Knop-Sahi recurrence.

Next we can show that the second of Knop-Sahi recurrences, the special case of the first recurrence in which $$\mu_{i+1} = 0$$, and $$E_0 = 1$$ completely characterize the nonsymmetric Macdonald polynomials. From the fact that they are eigenfunctions, we get that nonsymmetric Macdonald polynomials exist.

## Combinatorics

We begin by considering diagrams for the weak compositions $$\mu \in \mathbb{Z}_{\geq 0}^n$$ of length $$n$$ by drawing the $$i$$-th column as a length $$\mu_i$$ column aligned at the bottom. We denote this by $$dg(\mu)$$. We also consider augmented diagrams $$\widehat{dg}(\mu)$$ which is the usual diagram but adding a row of length length $$n$$ to the base.

We define the following sets of a cell $$u = (i, j) \in dg(\mu)$$:

• The leg is the set of cells directly above $$u$$. So all cells $$(i,j') \in dg(\mu)$$ such that $$j' > j$$.
• The left arm is the set of cells to the left of $$u$$ of equal or smaller height columns. So all cells $$(i', j) \in dg(\mu)$$ such that $$i' < i$$ and $$\mu_{i'} \leq \mu_i$$.
• The right arm is the set of cells to the right of $$u$$ in the row below $$u$$ of strictly smaller height columns. So all cells $$(i', j-1)$$ such that $$i' > i$$ and $$\mu_{i'} < \mu_i$$.
• The arm is the union of the left and right arms.

We now define the set of statistics on $$u$$. We begin by defining $$l(u) = \lvert \mathrm{leg}(u) \rvert = \mu_i - j$$ and $$a(u) = \lvert \mathrm{arm}(u) \rvert$$. Using these statistics, if $$\mu_i > \mu_{i+1}$$, we can reformulate our recursion by $E_{s_i(\mu)}(x; q, t) = \left( T_i + \dfrac{1 - t}{1 - q^{l(u)+1} t^{a(u)}} \right) E_{\mu}(x; q, t),$ where $$u = (i, \mu_{i+1} + 1)$$. We can also define an integral form for the non-symmetric Macdonald polynomials by $\mathcal{E}_{\mu}(x; q, t) = \prod_{u \in dg(\mu)} \left( 1 - q^{l(u)+1} t^{a(u)+1} \right) E_{\mu}(x; q, t).$

Our next statistics will be defined on fillings of the diagrams, which are just maps $$\sigma \colon dg(\mu) \to [n].$$ We can augment the filling by defining the map $$\widehat{\sigma} \colon \widehat{dg}(\mu) \to [n]$$ with $$\widehat{\sigma}\bigl( (0, j) \bigr) = j$$ and agrees with $$\sigma$$ everywhere else. We say two cells $$(a,b), (i,j)$$ attack each other if

• they are in the same row, i.e. $$b = j$$, or
• they are in consecutive rows and the box in the lower row is to the right of the one in the upper row, i.e. $$i < a$$ and $$b = j - 1$$.

We say an augmented filling is non-attacking if $$\widehat{\sigma}(u) \neq \widehat{\sigma}(v)$$ for all attacking pairs $$u,v \in \widehat{dg}(\mu)$$. We say a filling is non-attacking if its augmented filling is non-attacking.

Next let $$d(u) = (i, j-1)$$ be the box directly below $$u = (i, j)$$. A descent in the filling is a box $$u \in \widehat{\sigma}$$ such that $$d(u) \in \widehat{\sigma}$$ and $$\widehat{\sigma}(u) > \widehat{\sigma}(v).$$ We denote by $$Des(\widehat{\sigma})$$ as the set of descents of $$\widehat{\sigma}$$ and the major index is $maj(\widehat{\sigma}) = \sum_{u \in Des(\widehat{\sigma})} (l(u) + 1).$

We define the reading order of boxes in a diagram by reading row by row from right-to-left, top-to-bottom, i.e. $$(i, j) < (a, b)$$ if $$j > b$$, or if $$j = b$$ and $$i > a$$. An inversion of a filling $$\widehat{\sigma}$$ is a pair of attacking boxes $$u, v \in \widehat{dg}(\mu)$$ such that $$u < v$$ in the reading order and $$\widehat{\sigma}(u) > \widehat{\sigma}(v).$$ We denote this set by $$Inv(\widehat{\sigma})$$ and the inversion statistic by $inv(\widehat{\sigma}) = \lvert Inv(\widehat{\sigma}) \rvert - \lvert \{ i < j \mid \mu_i \leq \mu_j \} \rvert - \sum_{u \in Des(\widehat{\sigma})} a(u).$ We also define the co-inversion statistic by $coinv(\widehat{\sigma}) = -inv(\widehat{\sigma}) + \sum_{u \in dg(\mu)} a(u).$

Thus we can compute the non-symmetric Macdonald polynomials by $E_{\mu}(x; q, t) = \sum_{\sigma} x^{\sigma} q^{maj(\widehat{\sigma})} t^{coinv(\widehat{\sigma})} \prod_{\substack{ u \in dg(\mu) \\ \widehat{\sigma}(u) \neq \widehat{\sigma}(d(u))}} \dfrac{1 - t}{1 - q^{l(u)+1} t^{a(u)+1}},$ where we sum over all non-attacking fillings of $$dg(mu)$$ and $$x^{\sigma} = \prod_{u \in dg(\mu)} x_{\sigma(\mu)}.$$ We also have the integral form as $\mathcal{E}_{\mu}(x; q, t) = \sum_{\sigma} x^{\sigma} q^{maj(\widehat{\sigma})} t^{coinv(\widehat{\sigma})} \prod_{\substack{ u \in dg(\mu) \\ \widehat{\sigma}(u) \neq \widehat{\sigma}(d(u))}} 1 - q^{l(u)+1} t^{a(u)+1} \prod_{\substack{ u \in dg(\mu) \\ \widehat{\sigma}(u) \neq \widehat{\sigma}(d(u))}} (1 - t).$

Nonsymmetric Macdonald Polynomials is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.