Ensayos a realizar en suelos estabilizados

Beginning with Section 9.2, we assumed that our algebra

gwas simply laced. However, there are

minuscule weights in the non-simply laced type

B

andC

root systems. There is a single minuscule

weight for each type

C

system. There a dimension calculation shows that the corresponding Plücker

module

I

= 0, and so there are no Plücker relations for its flag variety. There is also a single

minuscule weight for each typeB

system. For this case we deduce results about its Plücker relations

from our results for a simply-laced type

D

case through the strategy of “diagram folding.” This

uses an embedding of a typeB

n−1

Lie algebra into one of typeD

n

. This strategy has been used for

example in [33, 34].

We describe this embedding concretely. Fixn≥3. LetV

be the 2n-dimensional complex vector

space with nondegenerate symmetric bilinear form

h·,·i

defined in Section 7.1, and consider the

orthogonal Lie algebrag:=o(V) which is simple of typeD

n

. Let

L⊂V

be the line spanned by

the vector

v

₀]

−v

₀[

and let

g

0

⊂g

be the subalgebra of endomorphisms that annihilate

L. Then

g

0

stabilizes the orthogonal complement

W

:=

L

⊥

; it is the hyperplane spanned by the vectors

{v

n

, . . . , v

1

, v

0]

+v

₀[

, v

₁

, . . . , v

n

}. It is simple to check that the restrictionh·,·i

W

of the bilinear form

to

W

is again nondegenerate. Every endomorphism of

W

that fixes

h·,·i

_W

is faithfully represented

in the restriction of the endomorphisms in

g

0

toW. Hence

g

0

is isomorphic to an orthogonal Lie

algebra for a 2n−1 dimensional vector space with nondegenerate symmetric bilinear form. Therefore

the subalgebra

g

0

⊂gis simple of type

B

n−1

, and so we rename it

g

B

. Label the typeB

n−1

and

typeD

n

Dynkin diagrams as in Figure 9.22. The subalgebra

h

B

:=h∩g

B

is a Cartan subalgebra

forg

B

. We have a map

h

∗

→h

∗B

induced by the inclusion

h

B

⊂hgiven by restriction of the domain.

In particular, the minuscule weightω

0

of

h

∗B

is the restriction

ω

]

|

hB

(or

ω

[

|

hB

) of a spin minuscule

weight of

h

∗

.

Let

λbe one of the two spin weights

ω

]

or

ω

[

of

g. LetP

:=P

λ

be the corresponding minuscule

poset, and construct Wildberger’s representation

V

J(P)

of

g. Define the unique

g-submodule

I

⊂Sym

2

(V

J(P)

) as for Problem 9.1. The subalgebra

g

B

⊂g

acts naturally on

V

J(P)

. Then the

g

B

submoduleV

B

:=U(g

B

).ℵ

of

V

J(P)

is an irreducible representation of

g

B

whose highest weight

λ

B

:=ω

0

is its unique minuscule weight. Now one can pose Problem 9.1 for the

g

B

-module

V

B

:

The

g

_B

-moduleSym

2

(V

B

) decomposes into a direct sum

U(g

B

).(ℵ)

2

⊕I

B

of

g

B

-submodules for a

unique submodule

I

B

. Find a spanning set (or basis) forI

B

.

Proposition 9.23.

The

g

_B

-modules

V

B

andV

J(P)

are equal. Moreover, the subspaces

I

and

I

B

of

Sym

2

_(V

J(P)

) =Sym

2

(V

B

)

are equal.

Proof.

We have that

V

B

⊆

V

J(P)

. The first assertion follows from the standard dimension fact

dim(V

B

) = 2

n−1

= dim(V

J(P)

). We have

U(g

B

).(ℵ)

2

⊆U(g).(ℵ)

2

. Since

g-modules are naturally

g

_B

-modules, the

g-module decomposition

Sym

2

_(V

J(P)

) =

U(g).(ℵ)

2

⊕I

is also a

g

B

-module de-

composition. Since the complementary

g

_B

-module

I

B

and

g-module

I

are each unique, we have

I

⊆I

B

.

We now show that dim(U(g

_B

).(ℵ)

2

_{) = dim(U(g).(ℵ)}

2

_{). We use the usual notation for the root}

system of

g. Let Φ

B

⊂h

∗_B

denote the root system ofg

B

. Let

ρ

B

∈h

∗_B

denote the Weyl vector for

the root system Φ

B

. By the Weyl dimension formula, we have that dim(V

B

) =

Y α∈Φ+_B

h2λ

B

+ρ

B

, αi

hρ

B

, αi

and dim(V

J(P)

) =

Y α∈Φ+

h2λ+ρ, αi

hρ, αi

. We takeλ

=

ω

]

. For roots

α

∈

Φ

B

(resp.

α

∈

Φ) with no

α0

(resp.

α

]

) component, we have that

h2λ

B

+ρ

B

, αi

=

hρ

B

, αi

(resp. without the subscript

B) and so the corresponding factor in the Weyl dimension formula above is 1. The remaining

roots in Φ

_B

are the short roots

{α

0

+

α

1

+

· · ·+α

i

}

for 0

≤

i

≤

n−2 and the long roots

{2α

0

+ 2α

1

+· · ·+ 2α

i

+α

i+1

+· · ·+α

j

}for 0≤i < j

≤n−2. The remaining roots in Φ are the

roots

{α

]

+α

1

+· · ·+α

i

}for 0≤i≤n−2 and

{α

]

+α

[

+ 2α

1

+· · ·+ 2α

i

+α

i+1

+· · ·+α

j

}

for

0≤i < j

≤n−2. These two sets of roots match up in the obvious fashion from Φ to Φ

_B

when

restricting the domain toh

B

. By considering short and long roots of Φ

B

separately, it is easy to see

that for matching rootsα∈Φ

B

andβ

∈Φ as above we have

hρ

B

, αi=hρ, βi. For both the short

and long roots

α∈Φ

B

above, we have that

h2λ

B

, αi= 2. Similarly for the rootsβ

∈Φ above, we

have thath2λ, βi= 2. Hence, all of the corresponding factors in the Weyl dimension formulae are

equal.

From dim(U(g

_B

).(ℵ)

2

_{) = dim(U(g).(ℵ)}

2

_{), it follows that dim(I}

B

) = dim(I). Therefore

I

B

=I

as vector spaces.

This proposition allows us to obtain some Plücker relations for

g

_B

by first applying Section 9.3

with the simply laced algebra

gto obtain extreme weight relations inI, and then recognizing those

as Plücker relations in

I

B

. From this we obtain the final ingredient for Theorem 9.5:

Corollary 9.24.

Each extreme weight Plücker relation for

g

_B

is the standard straightening law on

a double-tailed diamond sublattice ofL

λB

. Moreover, Corollary 9.10 also holds here.

Proof.

We claim that the lattices

L

λ

and

L

λB

are equal. We use

to denote the usual order on

h

∗

R

and

B

to denote the usual order on (h

B

)

∗

R

. The weights ofL

λB

are merely the weights of

L

λ

when restricted to

h

_B

. We claim thatµν

inL

λ

if and only ifµ|

gB

B

ν|

gB

in

L

λB

. Indeed, the

restriction of the simple roots of Φ to

g

_B

are again simple roots of Φ

_B

:

α

]

7→

α

0

, α

[

7→α

0

, α

1

7→

α1, . . . , α

n−2

7→

α

n−2

. It is then straightforward to see that any positive root of Φ

+

restricts to

a positive root of Φ

+_B

, and that every root of Φ

+_B

is the restriction of some root of Φ

+

. Since the

orders,

B

are defined in terms of positive roots, we have

L

λ

∼=L

λB

as lattices.

Then the Plücker relations forg

B

obtained by Proposition 9.23 retain in

L

λB

whatever order

structure they may have in

L

λ

. In particular, each extreme weight Plücker relation is again the

standard straightening law on a double-tailed diamond sublattice of

L

λB

.

Note that since the restriction of weights gives

L

λ

∼=

L

λB

as lattices it also gives an order

isomorphism of the uncolored minuscule posets of meet irreducible weightsP

λ

∼=P

λB

. However the

coloring functions

κ

andκ

B

on

P

λ

andP

λB

are not the same: elementsx∈P

λ

with eitherκ(x) =]

orκ(x) =[

now have the

B-color

κ

B

(x) = 0. The coloringκ

B

of

P

λB

was not used in the proof of

the proposition or its corollary.

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 115-120)