# Nonsymmetric Macdonald Polynomials

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### 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:

- Triangularity: \(E_{\mu} \in x^{\mu} + \mathbb{Q}(q, t)\{ x^{\lambda} \mid \lambda < \mu \}.\)
- 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 \[ \mathcal{E}_{\mu}(x; q, t) = \sum_{\sigma} x^{\sigma} q^{maj(\widehat{\sigma})} t^{coinv(\widehat{\sigma})} \prod_{\substack{ u \in dg(\mu) \\ \widehat{\sigma}(u) \neq \widehat{\sigma}(d(u))}} 1 - q^{l(u)+1} t^{a(u)+1} \prod_{\substack{ u \in dg(\mu) \\ \widehat{\sigma}(u) \neq \widehat{\sigma}(d(u))}} (1 - t). \]