The symmetric functions catalog

An overview of symmetric functions and related topics


Macdonald E polynomials

The non-symmetric Macdonald polynomials were introduced in [Opd95], [Mac96a] and [Che95b]. The non-symmetric Macdonald polynomials are closely related with affine root systems, and the double affine Hecke algebra. Their definition is rather indirect, and does not give an efficient way of computing these non-symmetric polynomials.

In [HHL08], J. Haglund, M. Haiman and N. Loehr found an explicit formula for the non-symmetric Macdonald polynomials in type $A,$ as a sum over non-attacking fillings. This model is the basis for the permuted basement Macdonald polynomials, and we refer to that page for definitions.

Later, an alternative combinatorial formula using alcove walks was proved by A. Ram and M. Yip [RY11], which works for all Lie types. At $q=t=0,$ their model reduces to the Littelmann path model.

In [BW19], an integrable vertex model is used to give an alternative formula for non-symmetric Macdonald polynomials.

See the book [Hag07] for more background on the type $A$ non-symmetric Macdonald polynomials. My personal research with non-symmetric Macdonald polynomials and further generalizations can be found in [Ale19aAS17aAS19a].

A model using multiline queues is introduced in [CMW18], and further explored in [CHMMW19a].

Monk's rule

W. Baratta [Bar09Bar11] give Monk type rules (terminology from Schubert calculus) for products of the form

\[ x_j \macdonaldE_\mu(x_1,\dotsc,x_n;q,t), \quad \elementaryE_1(\xvec) \cdot \macdonaldE_\mu(x_1,\dotsc,x_n;q,t), \quad \elementaryE_{r}(\xvec) \cdot\macdonaldE_\mu(x_1,\dotsc,x_n;q,t), \]

expanded again in the $\{\macdonaldE_\alpha\}$ basis. Baratta uses interpolation Macdonald polynomials in the proof.

In [HR22], the authors prove Monk type formulas for

\[ x_j \macdonaldE_\mu, \qquad (x_1+\dotsb + x_j) \macdonaldE_\mu, \qquad x_j^{-1} \macdonaldE_\mu, \qquad (x_j^{-1}+\dotsb + x^{-1}_j) \macdonaldE_\mu. \]

Halverson–Ram uses a different method based on intertwiners.

Non-symmetric q-Whittaker polynomials

The non-symmetric $q$-Whittaker polynomials are obtained by letting $t=0$ in $\macdonaldE_\mu(\xvec;q,t).$ The result can be though of as a non-symmetric analogue of the transformed Hall–Littlewood polynomials, and generalize the $q$-Whittaker functions.

The polynomials $\macdonaldE_\mu(\xvec;q,0)$ expand positively in key polynomials, and the coefficients are Kostka–Foulkes polynomials, see [AS17aAss18aAG18].

Models using quantum alcove walks and quantum Lakshmibai–Seshadri path are considered in [LNSSS17].