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\author[S. Mason]{S. Mason}
\address{Department of Mathematics, Davidson College}



\email{samason@davidson.edu}
\urladdr{http://www.davidson.edu/math/mason}

\title[Demazure atoms]{Comment on `A Decomposition of Schur Functions and an Analogue of the Robinson-Schensted-Knuth Algorithm'}

\newcommand{\C}{\mathbb{C}}


\begin{document}

\maketitle

The purpose of this comment is to clarify the connections between Demazure characters and the objects studied in this work.  The nonsymmetric Macdonald polynomials introduced by Macdonald \cite{Mac} and studied by Cherednick \cite{Che} are denoted by $E_{\alpha} (X; q,t)$, where $\alpha$ is a weak composition and $X=(x_1,x_2, \hdots)$.  The Demazure characters introduced by Demazure in \cite{Dem} and studied by Ion~\cite{Ion}, Joseph~\cite{Jos}, and Sanderson~\cite{San} are the specializations $E_{\alpha}(X;0,0)$.  

Marshall \cite{Mar} works with a variation of the above nonsymmetric polynomials obtained by reversing  the indexing composition, reversing the variables, and replacing $q$ and $t$ by $q^{-1}$ and $t^{-1}$ respectively.  These nonsymmetric polynomials, denoted $\hat{E}_{\alpha}(X;q,t)$, can therefore be written as $\hat{E}_{\alpha}(x_1,x_2,\hdots;q,t)=E_{{\rm reverse}(\alpha)}(\hdots, x_2, x_1; q^{-1},t^{-1})$.  It is these polynomials that we specialize to obtain the polynomials explored in this paper.  In fact, the specializations of the $\hat{E}_{\alpha}(X;q,t)$ to $q=t=0$ are equivalent to the second family of Demazure characters, often called ``standard bases" or ``Demazure atoms", introduced by Lascoux and Sch\"utzenberger in \cite{LS90} and studied by Lascoux in~\cite{Las}.  Please see \cite{M06} for a combinatorial proof of this equivalence.

We provide the following short table for the partition $\lambda=(2,1,0)$ to illustrate the distinction between $E_{\alpha}(X;0,0)$ and $\hat{E}_{\alpha}(X;0,0)$.

\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Composition $\alpha$ & $E_{\alpha}(X;0,0)$ & $\hat{E}_{\alpha}(X;0,0)$ \\
\hline
$(2,1,0)$ & $x_1^2x_2$ & $x_1^2 x_2$\\
\hline
$(2,0,1)$ & $x_1^2x_2 + x_1^2x_3$ & $x_1^2 x_3$\\
\hline
$(1,2,0)$ & $x_1^2x_2+x_1x_2^2$ & $x_1 x_2^2$\\
\hline
$(1,0,2)$ & $x_1^2x_2 + x_1x_2^2 + x_1^2x_3 + x_1 x_2 x_3 + x_1 x_3^2$ & $x_1 x_2 x_3 + x_1 x_3^2$\\
\hline
$(0,2,1)$ & $x_1^2x_2 + x_1x_2^2 + x_1^2 x_3 + x_1 x_2 x_3 + x_2^2 x_3$ & $x_1 x_2 x_3 + x_2^2 x_3$\\
\hline
$(0,1,2)$ & $x_1^2x_2 + x_1x_2^2 + x_1^2 x_3 + 2x_1 x_2 x_3 + x_2^2 x_3 + x_1x_3^2 + x_2 x_3^2$ & $x_2 x_3^2$\\
\hline
\end{tabular}
\end{center}


\bibliographystyle{amsalpha}
\begin{thebibliography}{A}

\bibitem{Che}
Cherednik, I,.  Nonsymmetric Macdonald polynomials, {\it Math. Res. Notices}, {\bf 10} (1995), pp. 483--515.

\bibitem{Dem}
Demazure, M.,
  D\'{e}singularisation des vari\'{e}t\'{e}s de Schubert,
  {\it Ann. E. N. S.},
  6 (1974),
  163--172.

\bibitem{Ion}
Ion, Bogdan,
Nonsymmetric {M}acdonald polynomials and {D}emazure characters,
{\it Duke Mathematical Journal},
116 (2003),
299--318

\bibitem{Jos}
Joseph, A.,
     On the {D}emazure character formula,
   {\it Ann. Sci. \'Ecole Norm. Sup.} (4),
  18:3 (1985),
    389--419.
    
\bibitem{LS90}
Lascoux, A., and Sch{\"u}tzenberger, M.-P.,
      Keys and Standard Bases,
      {\it Invariant Theory and Tableaux, IMA Volumes in Math and its Applications} (D. Stanton, ED.),
      Southend on Sea, UK,
      19 (1990),
      125--144.
      
      \bibitem{Las}
      A. Lascoux, Double Crystal graphs,
Studies in Memory of Issai Schur, {\it Progress In Math.}  210,
  Birkha\"user (2003) 95--114.
      
\bibitem{Mac}
Macdonald, I. G., {\it Affine Hecke algebras and orthogonal polynomials}, Ast\'{e}risque {\bf 237} (1996), pp.189--207, S\'{e}minaire Bourbaki 1994/95, Exp. no. 797.

\bibitem{Mar}
Dan Marshall, Symmetric and nonsymmetric {M}acdonald polynomials. {O}n combinatorics and statistical mechanics, {\it Ann. Comb.} \textbf{3} (1999), no.~2-4, 385--415.

\bibitem{M06}
Mason, S.,
   An explicit construction of type A Demazure Atoms,  to appear in {\it J. Algebraic Combinatorics}.
   
   \bibitem{San}
   Sanderson, Y., On the Connection between Macdonald polynomials and Demazure
    characters, {\it J. Algebraic Combin.} 11 (2000), no.3, 269-275.
    
\end{thebibliography}

\end{document}

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