Colored d-complete Posets Associated to the Weyl Group Orbits through Certain Weights for Multiply-laced Simple Lie Algebras, and Multiple Hook Removing Game Related to them

概要

The notion of a d-complete poset was introduced by Robert A. Proctor ([15, 16]). A
d-complete poset is a finite poset which satisfies some local conditions described in terms
of double-tailed diamonds (see Section 2.3), and can be regarded as extensions of Young
diagrams and shifted Young diagrams, having similar properties to the hook length property ([17]) and the jeu de taquin property ([18]) for Young diagrams. So, it is natural to
expect that d-complete posets play important roles in the combinatorial representation
theory as Young diagrams and shifted Young diagrams do.
We recall the fundamental relation between d-complete posets and finite-dimensional
simple Lie algebras (for the details, see Section 3.2). Let g be a simply-laced finitedimensional simple Lie algebra, with I the index set of simple roots. Let W = hsi | i ∈ Ii
be the Weyl group, where si is the simple reflection corresponding to i ∈ I. Let λ be a
dominant integral weight of g, and set Wλ := {w ∈ W | wλ = λ}. We know that each
coset in W/Wλ has a unique element whose length is minimal among the elements in
the coset; we regard W/Wλ as a subset of W by taking the complete system of these
“minimal-length coset representatives” for the cosets in W/Wλ . Let ≤s (resp., ≤w ) be
the partial order on W λ corresponding to the Bruhat order (resp., weak Bruhat order)
∼
on W/Wλ ⊂ W under the canonical map W λ → W/Wλ ⊂ W . If λ is minuscule (in this
case, ≤s is identical to ≤w ), then there exists a connected self-dual d-complete poset
(Pλ , ≤) such that (W λ, ≤s ) = (W λ, ≤w ) and (F(Pλ ), ⊆) are isomorphic as posets, where
F(Pλ ) is the set of order filters of Pλ ([15, Section 14]). Furthermore, using a unique map
κ : Pλ → I called the coloring, we construct an I-colored d-complete poset (Pλ , ≤, κ, I).
Then, there exists a unique order isomorphism f : W λ → F(Pλ ) satisfying the condition
that µ → si µ is a cover relation in W λ if and only if f (si µ) \ f (µ) consists of one element
x with κ(x) = i ([16, Proposition 9.1]). There are some important applications of these
results. For example, the problem counting the λ-minuscule elements in W is reduced
to the combinatorial problem counting the “standard tableaux” for the corresponding
d-complete posets ([22, Theorem 3.5]). Also, the “colored hook formula” for d-complete
posets obtained in [13] is a generalization of the famous hook length formula for Young
diagrams in terms of the reflections in the positive roots of g.
In this thesis, we study the relation between the Weyl group orbit through a dominant
integral weight for a multiply-laced finite-dimensional simple Lie algebra and the set
of order filters in a d-complete poset. To do this, we use the “folding” technique (see
Section 4.1). Assume that g is of type An , Dn , E6 , and let h be a Cartan subalgebra of
g. Let σ : I → I be a non-trivial automorphism of the Dynkin diagram of g; note that
σ canonically induces a Lie algebra automorphism of g such that σ(h) = h, and a linear
automorphism on the dual space h∗ of h. Then the fixed point subalgebra g(0) := {x ∈
g | σ(x) = x} is isomorphic to a multiply-laced finite-dimensional simple Lie algebra with
the set J of σ-orbits in I its index set of simple roots and h(0) := {h ∈ h | σ(h) = h}
˜ = h˜
its Cartan subalgebra. Let W
sp | p ∈ Ji ⊂ GL(h(0)∗ ) be the Weyl group of g(0).
ˆ := {w ∈ W | σwσ −1 = w} of W is isomorphic to W
˜.
We know that the subgroup W
ˆ
Let res : h∗ → h(0)∗ be the restriction map. The map res|W
ˆ λ gives a bijection W λ onto
˜ res(λ) for a dominant integral weight λ of g.
W
Now, let λ be a minuscule dominant integral weight of g, and (Pλ , ≤) the corresponding
d-complete poset mentioned above; recall the order isomorphism f : W λ → F(Pλ ). ...