On the Hypersurface Orbital Varieties of sl(N,C)

Benlolo, Elise; Sanderson, Yasmine B.

doi:10.1006/jabr.2001.8903

Cited by 6 publications

(26 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The leading term, gr M s , of its restriction is the Benlolo-Sanderson invariant. As noted in [3] its degree d(gr M s ) is given by…”

Section: Weyl Group Elementsmentioning

confidence: 98%

“…Let us just compute the excluded co-ordinates arising from the neighbouring columns of height 2. The first set occurs in column 6 in rows (1,2,3). The second set occurs in columns 10, 11 in rows 4, 5, 7 illustrating the gap and there being two columns, since c 5 − s = 2.…”

Section: 7mentioning

confidence: 99%

“…This finally excludes the use of the gated line ℓ 5,6 . We may remark that the composite lines joining the columns of height 2 can be defined by the sequence of boxes they pass through, Namely (2, 5, 7, 10) and (3,4,6,8,9). Notice that the composite lines exchange the entries of the last column compared to step 2.…”

Section: Proof Of P Imentioning

confidence: 99%

“…In type A, the invariant with respect to a given pair of neighbouring columns is the appropriate Benlolo-Sanderson invariant [3]. In [9,Sect.…”

Section: Introductionmentioning

confidence: 99%

“…Factorisation. In the notation of 1.2 suppose that C 1 , C k are neighbouring columns of height s. Then the weight of the Benlolo-Sanderson invariant is the Kostant weight ̟ s − w 0 ̟ s , [3,VIII], [16, 2.20], as is obvious from its description in [9, 3.6.3]. In the Bourbaki notation [2], set ε u,v = v i=u ε i .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Parabolic Adjoint Action, Weierstrass Sections and Components of the Nilfibre in Type $A$

Fittouhi,

Joseph

2021

Preprint

View full text Add to dashboard Cite

This work is a continuation of [Y. Fittouhi and A. Joseph, Weierstrass Sections for Parabolic adjoint action in type A].Let G be an irreducible simple algebraic group and B a Borel subgroup of G. Let n be the Lie algebra of the nilradical of B. Consider an irreducible subgroup P of G containing B. Let P ′ be the derived group of P . Let m be the Lie algebra of the nilradical of P .A theorem of Richardson asserts that the algebra C[m] P ′ of P semi-invariants is multiplicity-free. It is hence a polynomial algebra on generators which can be taken to be those weight vectors which are irreducible as polynomials.A linear subvariety e + V such that the restriction map induces an isomorphism of C[m] P ′ onto C[e + V ] is called a Weierstrass section for the action of P ′ on m.Here in type A such a section is constructed, but in better form than that given in Sect. 4, loc cit. Yet the main difference is a complete change of emphasis from the construction of a Weierstrass section, to its application.Let N be the nilfibre relative to this action. From the construction of a Weierstrass section e + V , it is shown that e ∈ N . Then P.e is contained in a unique irreducible component C of N .The structure of e + V is used to give a rather explicit description of C as a B saturation set, that is of the form B.u, where u is a subalgebra of n . This algebra is not necessarily complemented by a subalgebra in n and so B.u is not necessarily an orbital variety closure (hence Lagrangian) but it can be.

show abstract

“…The leading term, gr M s , of its restriction is the Benlolo-Sanderson invariant. As noted in [3] its degree d(gr M s ) is given by…”

Section: Weyl Group Elementsmentioning

confidence: 98%

Section: 7mentioning

confidence: 99%

Section: Proof Of P Imentioning

confidence: 99%

“…In type A, the invariant with respect to a given pair of neighbouring columns is the appropriate Benlolo-Sanderson invariant [3]. In [9,Sect.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Parabolic Adjoint Action, Weierstrass Sections and Components of the Nilfibre in Type $A$

Fittouhi,

Joseph

2021

Preprint

View full text Add to dashboard Cite

show abstract

Quantization of Hypersurface Orbital Varieties in sl n

Joseph

Melnikov

2003

The Orbit Method in Geometry and Physics

View full text Add to dashboard Cite

The Canonical component of the nilfibre for parabolic adjoint action in type A

Fittouhi,

Joseph

2024

J Algebr Comb

View full text Add to dashboard Cite

This work is a continuation of [Y. Fittouhi and A. Joseph, Parabolic adjoint action, Weierstrass Sections and components of the nilfibre in type A]. Let P be a parabolic subgroup of an irreducible simple algebraic group G. Let $$P'$$ P ′ be the derived group of P, and let $${\mathfrak {m}}$$ m be the Lie algebra of the nilradical of P. A theorem of Richardson implies that the subalgebra $${\mathbb {C}}[{\mathfrak {m}}]^{P'}$$ C [ m ] P ′ , spanned by the P semi-invariants in $${\mathbb {C}}[{\mathfrak {m}}]$$ C [ m ] , is polynomial. A linear subvariety $$e+V$$ e + V of $${\mathfrak {m}}$$ m is called a Weierstrass section for the action of $$P'$$ P ′ on $${\mathfrak {m}}$$ m , if the restriction map induces an isomorphism of $${\mathbb {C}}[{\mathfrak {m}}]^{P'}$$ C [ m ] P ′ onto $${\mathbb {C}}[e+V]$$ C [ e + V ] . Thus, a Weierstrass section can exist only if the latter is polynomial, but even when this holds its existence is far from assured. Let $${\mathscr {N}}$$ N be zero locus of the augmentation $${\mathbb {C}}[{\mathfrak {m}}]^{P'}_+$$ C [ m ] + P ′ . It is called the nilfibre relative to this action. Suppose $$G=\textrm{SL}(n,{\mathbb {C}})$$ G = SL ( n , C ) , and let P be a parabolic subgroup. In [Y. Fittouhi and A. Joseph, loc. cit.], the existence of a Weierstrass section $$e+V$$ e + V in $${\mathfrak {m}}$$ m was established by a general combinatorial construction. Notably, $$e \in {\mathscr {N}}$$ e ∈ N and is a sum of root vectors with linearly independent roots. The Weierstrass section $$e+V$$ e + V looks very different for different choices of parabolics but nevertheless has a uniform construction and exists in all cases. It is called the “canonical Weierstrass section”. Through [Y. Fittouhi and A. Joseph, loc. cit. Prop. 6.9.2, Cor. 6.9.8], there is always a “canonical” component $${\mathscr {N}}^e$$ N e of $${\mathscr {N}}$$ N containing e. It was announced in [Y. Fittouhi and A. Joseph, loc. cit., Prop. 6.10.4] that one may augment e to an element $$e_\textrm{VS}$$ e VS by adjoining root vectors. Then the linear span $$E_\textrm{VS}$$ E VS of these root vectors lies in $$\mathscr {N}^e$$ N e and its closure is just $${\mathscr {N}}^e$$ N e . Yet, this same result shows that $${\mathscr {N}}^e$$ N e need not admit a dense P orbit [Y. Fittouhi and A. Joseph, loc. cit., Lemma 6.10.7]. For the above [Y. Fittouhi and A. Joseph, loc. cit., Theorem 6.10.3] was needed. However, this theorem was only verified in the special case needed to obtain the example showing that $${\mathscr {N}}^e$$ N e may fail to admit a dense P orbit. Here a general proof is given (Theorem 4.4.5). Finally, a map from compositions to the set of distinct non-negative integers is defined. Its image is shown to determine the canonical Weierstrass section. One may anticipate that the remaining components of $${\mathscr {N}}$$ N can be similarly described. However, this is a long story and will be postponed for a subsequent paper. These results should form a template for general type.

show abstract

On the Hypersurface Orbital Varieties of sl(N,C)

Cited by 6 publications

References 11 publications

Parabolic Adjoint Action, Weierstrass Sections and Components of the Nilfibre in Type $A$

Parabolic Adjoint Action, Weierstrass Sections and Components of the Nilfibre in Type $A$

Quantization of Hypersurface Orbital Varieties in sl n

The Canonical component of the nilfibre for parabolic adjoint action in type A

Contact Info

Product

Resources

About