Constructions of representations of o(2n+1,C) that imply Molev and Reiner–Stanton lattices are strongly Sperner

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“…Molev's bases in [14][15][16] Open Open Open Open basis "restricts irreducibly" (see Section 3) under the action of a Lie subalgebra obtained by removing the generators corresponding to a certain node of the Dynkin diagram. The distributive lattice bases obtained from [6] for the irreducible representations G 2 (kω 1 ) do not restrict irreducibly under the action of any Lie subalgebra obtained in this way; in recent collaboration with the co-authors of that paper we have been able to show that these bases are solitary and edge-minimal. In Section 3 of this paper we develop tools which allow us to confirm in Sections 4, 5, and 6 the entries in the first three rows of Table 1.…”

“…In Section 3 of this paper we develop tools which allow us to confirm in Sections 4, 5, and 6 the entries in the first three rows of Table 1. In [4][5][6], we use these same techniques to confirm the results of rows four through seven. The familiar Gelfand-Tsetlin bases of [7] for the irreducible representations of sl(n + 1, C) are known to possess the distributive lattice property (e.g.…”

“…In this paper we show that any supporting graph P is the Hasse diagram for the poset defined to be the transitive closure of the directed graph P. We apply Proctor's sl (2, C) version [17] of a technique of Stanley and Griggs to see that any connected poset arising in this way is rank symmetric, rank unimodal, and strongly Sperner. This method is used in [6] to confirm the conjecture of Reiner and Stanton that certain lattices shown to be rank symmetric and unimodal in [20] are also strongly Sperner. Second, this paper provides tools which give some direction for producing a weight basis for a given representation and for identifying the coefficients for the actions of generators on the elements of the basis.…”

“…Second, this paper provides tools which give some direction for producing a weight basis for a given representation and for identifying the coefficients for the actions of generators on the elements of the basis. These or related techniques are used in [3][4][5][6], and Section 6 below to explicitly construct new families of weight bases. Prior to our earliest results (from 1995), there was only one construction (from 1950) of a family of representations for which the actions of the Chevalley generators on the elements of a weight basis were explicitly given [7].…”

“…These "extremal" properties are defined in terms of supporting graphs and appear to be possessed only by rare weight bases. In this paper and the sequels [4][5][6], we study particular families of weight bases in terms of these properties.…”