Acción de la cal sobre los suelos arcillosos

This section is an informal discussion of the extreme relations in type

Awithout formal proofs.

We relate the combinatorial notions for minuscule posets in this case to established combinatorial

notions for Young diagrams. Interestingly, these combinatorial notions of “content” and “rim hooks”

usually arise from the representation theory of the symmetric group. Use the labelling of typeA

n−1

roots and weights from Section 5.2. Fix a minuscule weightλ=ω

j

.

Rotate the Hasse diagram for the poset

P

:=a

n−1

(j) as pictured in Figure 8.2 clockwise by

45

◦

to obtain a rectangular array of dots with

j

rows and

n−j

columns. A rotated filter of the

poset is a left justified subarray of dots such that the number of dots in each of its rows is weakly

decreasing. Fix a filter

J

⊂P. Let

µ1, µ2, . . . , µ

j

be the number of dots in the 1st, 2nd,. . .,

jth

rows of the rotated depiction of

J. Then

µ

:= (µ

1

, µ

2

, . . . , µ

n−j

) is a partition. The dots in the

rotated depiction of

J

is the

Ferrer’s diagram

of

µ. If we replace each dot with a box, then we

obtain the Young diagram ofµ. This rotation process is a bijection from the filters ofP

(and hence

a basis of the

g-moduleV

J(P)

) to the Young diagrams fitting inside aj×n−j

rectangle.

In this depiction of filters, the boxes represent elements of

P. It is not difficult to verify the

following description of the coloring functionκ

onP. Each diagonal of a Young diagram is assigned

a distinct color. The diagonal consisting of the location (j,1) at the southwest corner of thej×n−j

rectangle is assigned color 1. The diagonal of locations (j−1,1) and (j,2) is assigned color 2,

and so on. In particular, the main diagonal of locations (1,1),(2,2), . . . ,(j, j) is assigned color

j.

This coloring of the locations is similar to the usual notion of the

content

of a location in a Young

diagram, except that contents on such diagrams range from 1−j

ton−1−j

instead of 1 to

n−1.

We now show that

α-layers for rootsα∈Φ

+

are simply “rim hooks” in this formulation.

Definition 9.25.

A

rim hook

is a connected subset of boxes in the

j×n−j

array that does not

have two boxes on the same diagonal.

It is possible to show thatα-layers must be connected. In typeA

n−1

, the simple root expansion of a

rootα∈Φ

+

isα=α

i

+α

i+1

+· · ·+α

j

for some 1≤i≤j

≤n−1. Fix such an

α, and suppose that

R⊂P

is anα-layer. Then

R

has a single element of each of the colorsi, i+ 1, . . . , j. Hence when

rotated, the subsetR

does not have two boxes on the same diagonal. Therefore the rotated depiction

ofRis a rim hook. On the other hand, suppose thatH

is a rim hook in thej×n−jarray. It clearly

corresponds to a convex subset of

P. Leti

denote the color of its southwesternmost box, and letj

denote the color of its northeasternmost box. SinceH

is connected, it must have a box in each of

the diagonals of color

i+ 1, . . . , j−1. SinceH

does not have two boxes on the same diagonal, it has

exactly one box of each of these colors. Therefore we have

P

and thatH

is anα-layer.

According to Theorem 9.5, the extreme weight Plücker relations are standard straightening laws

on double-tailed diamond sublattices of

L

λ

. As we saw in Section 9.2, in typeA

these double-tailed

diamond lattices are isomorphic to the one that appears in the modelD3

case. Hence there are

only three terms in these straightening laws: the product of the incomparable pair, the product of

their meet and join, and one other standard monomial. It is simple to describe this last standard

monomial in typeA

without using the determinantal Plücker relations from Section 3.1. First we

need some preliminary definitions:

Definition 9.26.

Letα, β∈Φ

+

and writeα=α

i

+α

i+1

+· · ·+α

j

andβ

=α

k

+α

k+1

+· · ·+α

`

. If

j < k−1 or

` < i−1, thenα

andβ

are said to beseparated. In that case, without loss of generality

suppose

j < k

and define theirbridge root

γ(α, β) to beγ(α, β) :=α

j+1

+α

j+2

+· · ·+α

k−1

.

Recall that for any root

α∈Φ

+

, we defined the actions of root vectors

e

α

, f

α

of

g

onV

J(P)

in

Section 8.2.

Proposition 9.27.

Let

J, K∈J(P)

be incomparable filters such that

J−(J∨K)

and

K−(J∨K)

are rim hooks. Then these rim hooks are root layers for separated rootsα, β

∈Φ

+

and we have the

type

A

straightening law:

J K= (J ∨ K)(J ∧ K)−(e

_γ(α,β)

.[J ∨ K])(f

_γ(α,β)

.[J ∧ K]).

The filters of the two vectors in the final standard monomial are obtained from the meet and join by

transferring a rim hook of color

γ(α, β)

from the join filter to the meet filter.

In Section 9.1, we claimed that the Plücker moduleI

is minuscule in typeA

n−1

for the minuscule

weights

ω2

andω

n−2

. In these cases, the extreme weight Plücker relations given by Theorem 9.5

and described in detail by Proposition 9.27 above form a basis of

I. We now confirm that claim.

First define

ω0

andω

n

to be the trivial weight 0. For

λ=ω

j

we can see from the diagrams of the

top incomparable pairZ

]

_{, Z}

[

_{that the highest weightη}

_{of the submoduleI}

_isω

j−2

+ω

j+2

. So when

j= 2, this weight isω

4

. And when

j=n−2, this weight isω

n−4

. In both cases, a dimension count

In document Aprovechamiento de escorias blancas (LFS) y negras (EAFS) de acería eléctrica en la estabilización de suelos y en capas de firmes de caminos rurales (página 120-123)