Differential posets and Smith normal forms. (English) Zbl 1228.05096

Miller, Alexander; Reiner, Victor

Order 26, No. 3, 197-228 (2009).

Summary: We conjecture a strong property for the up and down maps \(U\) and \(D\) in an \(r\)-differential poset: \(DU + tI\) and \(UD + tI\) have Smith normal forms over \({\mathbb Z}[t]\). In particular, this would determine the integral structure of the maps \(U, D, UD, DU\), including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice \(\mathbf Y F\) studied by Okada and its \(r\)-differential generalizations \(Z(r)\), as well as verifying many of its consequences for Young’s lattice \(\mathbf Y\) and the \(r\)-differential Cartesian products \(Y ^{r }\).

MSC:

05B20	Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05A15	Exact enumeration problems, generating functions
05A17	Combinatorial aspects of partitions of integers
06A07	Combinatorics of partially ordered sets

Keywords:

invariant factors; Smith normal form; differential poset; dual graded graphs

References:

[1]	Bergeron, N., Lam, T., Li, H.: Combinatorial Hopf algebras and Towers of Algebras. arXiv:0710.3744 (2008) · Zbl 1281.16036
[2]	Fomin, S.: Duality of graded graphs. J. Algebr. Comb. 3, 357–404 (1994) · Zbl 0810.05005 · doi:10.1023/A:1022412010826
[3]	Fomin, S.: Schensted algorithms for dual graded graphs. J. Algebr. Comb. 4, 5–45 (1995) · Zbl 0817.05077 · doi:10.1023/A:1022404807578
[4]	Fulton, W., Harris, J.: Representation Theory, A First Course. Springer GTM 129. Springer, New York (1991) · Zbl 0744.22001
[5]	Kuperberg, G.: Kasteleyn cokernels. Electron. J. Comb. 9, 30 pp (2002) · Zbl 1006.05005
[6]	Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford (1979) · Zbl 0487.20007
[7]	Nzeutchap, J.: Dual graded graphs and Fomin’s r-correspondences associated to the Hopf algebras of planar binary trees, quasi-symmetric functions and noncommutative symmetric functions. In: FPSAC (2006)
[8]	Nzeutchap, J.: Binary search tree insertion, the hypoplactic insertion, and dual graded graphs. arXiv:0705.2689 (2007)
[9]	Okada, S.: Wreath products by the symmetric groups and product posets of Young’s lattices. J. Comb. Theory, Ser. A 55, 14–32 (1990) · Zbl 0707.05062 · doi:10.1016/0097-3165(90)90044-W
[10]	Stanley, R.P.: Differential posets. J. Am. Math. Soc. 1, 919–961 (1988) · Zbl 0658.05006 · doi:10.1090/S0894-0347-1988-0941434-9
[11]	Stanley, R.P.: Variations on differential posets. In: Stanton, D. (ed.) Invariant Theory and Tableaux, IMA Vol. Math. Appl., no. 19, pp. 145–165. Springer, New York (1988)
[12]	Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999) · Zbl 0928.05001
[13]	Stembridge, J.: On the eigenvalues of representations of reflection groups and wreath products. Pac. J. Math. 140, 353–396 (1989) · Zbl 0641.20011
[14]	Zelevinsky, A.V.: Representations of finite classical groups. A Hopf algebra approach. In: Lecture Notes in Mathematics, vol. 869. Springer, Berlin (1981) · Zbl 0465.20009