Abstract
We refine the classical Littlewood-Richardson rule in several different settings. We begin with a combinatorial rule for the product of a Demazure atom and a Schur function. Building on this, we also describe the product of a quasisymmetric Schur function and a Schur function as a positive sum of quasisymmetric Schur functions. Finally, we provide a combinatorial formula for the product of a Demazure character and a Schur function as a positive sum of Demazure characters. This last rule implies the classical Littlewood-Richardson rule for the multiplication of two Schur functions. Nous décrivons trois nouvelles règles de Littlewood-Richardson, et chaque nouvelle règle partage la vieille règle de Littlewood-Richardson. La première règle multiplie un atome de Demazure et une fonction de Schur. La deuxième multiplie une fonction de quasisymmetric-Schur et une fonction de Schur. La troisième multiplie un caractère de Demazure et une fonction de Schur. Cette dernière règle est une description de la vieille règle de Littlewood-Richardson.
Highlights
By letting q = t = 0, we obtain a new combinatorial formula for the Eα(X; 0; 0), which are known [13] to be certain B-module characters studied by Demazure, commonly referred to as Demazure characters
The combinatorics of the case q = t = 0 of these polynomials, i.e. Eαn,...,α1, was investigated in [10], [11], including a direct combinatorial proof that they are the same as polynomials introduced by Lascoux and Schutzenberger [7] in connection with the study of Schubert polynomials, which they called standard bases, and which equal the characters of quotients of Demazure modules
It is known that the Demazure character is a positive sum of Demazure atoms, and that the Schubert polynomial is a positive sum of Demazure characters
Summary
In [5], Haglund, Haiman, and Loehr obtained a new combinatorial formula for the type A nonsymmetric Macdonald polynomial Eα(x1, . . . , xn; q, t) first introduced by Macdonald [8], where α is a (weak) composition into n nonnegative parts. Schur functions are special cases of both Demazure characters and Schubert polynomials, and the decomposition of a Schur function as a positive sum of Demazure atoms was proved directly in [10] using an extension of the RSK algorithm. It is well-known [2] that the Schur function sλ(x1, . The authors showed that if you multiply a quasisymmetrc Schur function by an elementary symmetric function ek(= s1k ) or a complete homogeneous symmetric function hk(= sk) this result can be expressed in a simple combinatorial way as a positive sum of quasisymmetric Schur functions From this rule the classical Pieri rule for multiplying a Schur function by an ek or hk can be derived. One can obtain the classical LR-rule from the rule for Demazure atoms by careful bookkeeping combined with the decomposition of Schur functions into atoms
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