Elementary Proofs of Identities for Schur Functions and Plane Partitions
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چکیده
By a plane partition, we mean a finite set, P , of lattice points with positive integer coefficients, {(i, j, k)} ⊆ N, with the property that if (r, s, t) ∈ P and 1 ≤ i ≤ r, 1 ≤ j ≤ s, 1 ≤ k ≤ t, then (i, j, k) must also be in P . A plane partition is symmetric if (i, j, k) ∈ P if and only if (j, i, k) ∈ P . The height of stack (i, j) is the largest value of k for which there exists a point (i, j, k) in the plane partition. A plane partition is column strict if the height of stack (i, j) is strictly less than the height of stack (i− 1, j) whenever i ≥ 2 and (i, j, 1) is in the plane partition. Symmetric plane partitions were studied by P. A. MacMahon [12] who conjectured in 1898 that the generating function for symmetric plane partitions with 1 ≤ i, j ≤ n and 1 ≤ k ≤ m is
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Abstract. Major Percy A. MacMahon’s first paper on plane partitions [4] included a conjectured generating function for symmetric plane partitions. This conjecture was proven almost simultaneously by George Andrews and Ian Macdonald, Andrews using the machinery of basic hypergeometric series [1] and Macdonald employing his knowledge of symmetric functions [3]. The purpose of this paper is to sim...
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تاریخ انتشار 1998