Reduced Words and Plane Partitions

AbstractLet w0 be the element of maximal length in thesymmetric group Sn, and let Red(w0) bethe set of all reduced words for w0. We prove the identity $$\sum\limits_{(a_1 ,a_2 , \ldots ) \in Red(w_0 )} {(x + a_1 )(x + a_2 )} \cdots = \left( {_2^n } \right)!\prod\limits_{1 \leqslant i < j \leqslant n} {\frac{{2x + i + j - 1}}{{i + j - 1}}} ,$$ which generalizes Stanley's [20] formula forthe cardinality of Red(w0), and Macdonald's [11] formula $$\sum {a_1 a_2 \cdots = (_2^n )} !$$ .Our approach uses anobservation, based on a result by Wachs [21], that evaluation of certainspecializations of Schubert polynomials is essentially equivalent toenumeration of plane partitions whose parts are bounded from above. Thus,enumerative results for reduced words can be obtained from the correspondingstatements about plane partitions, and vice versa. In particular, identity(*) follows from Proctor's [14] formula for the number of planepartitions of a staircase shape, with bounded largest part.Similar results are obtained for other permutations and shapes;q-analogues are also given.

[1]  Robert A. Proctor.  New Symmetric Plane Partition Identities from Invariant Theory Work of De Concini and Procesi , 1990, Eur. J. Comb..

[2]  Itaru Terada,et al.  Young-diagrammatic methods for the representation theory of the classical groups of type Bn, Cn, Dn , 1987 .

[3]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[4]  R. King.  Weight multiplicities for the classical groups , 1976 .

[5]  P. M. Cohn.  ALGEBRES DE LIE SEMI‐SIMPLES COMPLEXES , 1968 .

[6]  I. G. MacDonald,et al.  Notes on Schubert polynomials , 1991 .

[7]  J. Shaw.  Combinatory Analysis , 1917, Nature.

[8]  Ira M. Gessel,et al.  Determinants, Paths, and Plane Partitions , 1989 .

[9]  Sergey Fomin,et al.  Schubert Polynomials and the Nilcoxeter Algebra , 1994 .

[10]  THE CLASSICAL GROUPS. SPECTRAL ANALYSIS OF THEIR FINITE-DIMENSIONAL REPRESENTATIONS , 1962 .

[11]  Sergey Fomin,et al.  Balanced Labellings and Schubert Polynomials , 1997, Eur. J. Comb..

[12]  Michelle L. Wachs,et al.  Flagged Schur Functions, Schubert Polynomials, and Symmetrizing Operators , 1985, J. Comb. Theory, Ser. A.

[13]  Richard P. Stanley,et al.  Some Combinatorial Properties of Schubert Polynomials , 1993 .

[14]  Robert A. Proctor.  Odd symplectic groups , 1988 .

[15]  Victor Reiner,et al.  Key Polynomials and a Flagged Littlewood-Richardson Rule , 1995, J. Comb. Theory A.

[16]  RICHARD P. STANLEY,et al.  On the Number of Reduced Decompositions of Elements of Coxeter Groups , 1984, Eur. J. Comb..

[17]  G. B. Mathews,et al.  Combinatory Analysis. Vol. II , 1915, The Mathematical Gazette.

[18]  Sergey Fomin,et al.  The Yang-Baxter equation, symmetric functions, and Schubert polynomials , 1996, Discret. Math..

[19]  Alain Lascoux,et al.  Polynômes de Schubert Une approche historique , 1995, Discret. Math..