Lie Theoretic Comments

Lie Theoretic Comments


Definition of lambda-Minuscule

Let g be a simply laced Kac-Moody algebra with Weyl group W. Let lambda be a highest weight for g and let w be an element of W. Then Dale Peterson defines w to be lambda-minuscule if there exists some reduced decomposition for w which is such that when it is applied to lambda, the application of each successive Coxeter generator results in the subtraction of just one positive simple root at each step.


Connection Between lambda-Minuscule and d-Complete

The definition of d-complete poset is given in terms of local structural properties.
The first main result of the paper 'Minuscule' states that a certain poset associated to a lambda-minuscule element will always be a distributive lattice. Combinatorialists have known for some time that any distributive lattice can be described in terms of a relatively small subset of its elements, the join-irreducible elements. The second main result of this paper states that a distributive lattice can arise from a lambda-minuscule element if and only if its poset of join irreducible elements is d-complete. So lambda-minuscule Weyl group elements are equivalent in a certain fashion to d-complete posets.


Basis Origin of Notion of d-Complete

In 1993 we noticed that the simplest cases of the Lakshmibai-Seshadri basis results, those for Demazure modules whose lowest weights were integer multiples of weights obtained by acting on lambda with a lambda-minuscule w, could be given purely combinatorial interpretations and proofs. This was done in the setting of pouring colored marbles into bins whose internal structures were described by the associated colored d-complete posets. We were also able to prove that d-complete posets were the only colored posets for which such basis results could be obtained for the Z-modules associated in a certain way to the colored poset.


Reduced Decompositions

Let w be a lambda-minuscule element of a simply laced Kac-Moody Weyl group. Then there exists a certain coloring of the elements of the associated d-complete poset P(w), where the colors correspond to the nodes of the Dynkin diagram for the Kac-Moody algebra. The ways of reading off the elements of the poset which are consistent with the partial order on the poset correspond to the reduced decompositions of w. Hence the number of reduced decompositions of w is equal to the number of order extensions of P(w). In joint work with Dale Peterson, we have obtained a hook product formula for this quantity which generalizes the Frame-Robinson-Thrall hook formula.

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