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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