{"title":"The Canonical component of the nilfibre for parabolic adjoint action in type A","authors":"","doi":"10.1007/s10801-023-01296-6","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>This work is a continuation of [Y. Fittouhi and A. Joseph, Parabolic adjoint action, Weierstrass Sections and components of the nilfibre in type <em>A</em>]. Let <em>P</em> be a parabolic subgroup of an irreducible simple algebraic group <em>G</em>. Let <span> <span>\\(P'\\)</span> </span> be the derived group of <em>P</em>, and let <span> <span>\\({\\mathfrak {m}}\\)</span> </span> be the Lie algebra of the nilradical of <em>P</em>. A theorem of Richardson implies that the subalgebra <span> <span>\\({\\mathbb {C}}[{\\mathfrak {m}}]^{P'}\\)</span> </span>, spanned by the <em>P</em> semi-invariants in <span> <span>\\({\\mathbb {C}}[{\\mathfrak {m}}]\\)</span> </span>, is polynomial. A linear subvariety <span> <span>\\(e+V\\)</span> </span> of <span> <span>\\({\\mathfrak {m}}\\)</span> </span> is called a Weierstrass section for the action of <span> <span>\\(P'\\)</span> </span> on <span> <span>\\({\\mathfrak {m}}\\)</span> </span>, if the restriction map induces an isomorphism of <span> <span>\\({\\mathbb {C}}[{\\mathfrak {m}}]^{P'}\\)</span> </span> onto <span> <span>\\({\\mathbb {C}}[e+V]\\)</span> </span>. Thus, a Weierstrass section can exist only if the latter is polynomial, but even when this holds its existence is far from assured. Let <span> <span>\\({\\mathscr {N}}\\)</span> </span> be zero locus of the augmentation <span> <span>\\({\\mathbb {C}}[{\\mathfrak {m}}]^{P'}_+\\)</span> </span>. It is called the nilfibre relative to this action. Suppose <span> <span>\\(G=\\textrm{SL}(n,{\\mathbb {C}})\\)</span> </span>, and let <em>P</em> be a parabolic subgroup. In [Y. Fittouhi and A. Joseph, loc. cit.], the existence of a Weierstrass section <span> <span>\\(e+V\\)</span> </span> in <span> <span>\\({\\mathfrak {m}}\\)</span> </span> was established by a general combinatorial construction. Notably, <span> <span>\\(e \\in {\\mathscr {N}}\\)</span> </span> and is a sum of root vectors with linearly independent roots. The Weierstrass section <span> <span>\\(e+V\\)</span> </span> 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 <span> <span>\\({\\mathscr {N}}^e\\)</span> </span> of <span> <span>\\({\\mathscr {N}}\\)</span> </span> containing <em>e</em>. It was announced in [Y. Fittouhi and A. Joseph, loc. cit., Prop. 6.10.4] that one may augment <em>e</em> to an element <span> <span>\\(e_\\textrm{VS}\\)</span> </span> by adjoining root vectors. Then the linear span <span> <span>\\(E_\\textrm{VS}\\)</span> </span> of these root vectors lies in <span> <span>\\(\\mathscr {N}^e\\)</span> </span> and its closure is just <span> <span>\\({\\mathscr {N}}^e\\)</span> </span>. Yet, this same result shows that <span> <span>\\({\\mathscr {N}}^e\\)</span> </span> need <em>not</em> admit a dense <em>P</em> 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 <span> <span>\\({\\mathscr {N}}^e\\)</span> </span> may fail to admit a dense <em>P</em> 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 <span> <span>\\({\\mathscr {N}}\\)</span> </span> 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.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"3 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10801-023-01296-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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 \(P'\) be the derived group of P, and let \({\mathfrak {m}}\) be the Lie algebra of the nilradical of P. A theorem of Richardson implies that the subalgebra \({\mathbb {C}}[{\mathfrak {m}}]^{P'}\), spanned by the P semi-invariants in \({\mathbb {C}}[{\mathfrak {m}}]\), is polynomial. A linear subvariety \(e+V\) of \({\mathfrak {m}}\) is called a Weierstrass section for the action of \(P'\) on \({\mathfrak {m}}\), if the restriction map induces an isomorphism of \({\mathbb {C}}[{\mathfrak {m}}]^{P'}\) onto \({\mathbb {C}}[e+V]\). Thus, a Weierstrass section can exist only if the latter is polynomial, but even when this holds its existence is far from assured. Let \({\mathscr {N}}\) be zero locus of the augmentation \({\mathbb {C}}[{\mathfrak {m}}]^{P'}_+\). It is called the nilfibre relative to this action. Suppose \(G=\textrm{SL}(n,{\mathbb {C}})\), and let P be a parabolic subgroup. In [Y. Fittouhi and A. Joseph, loc. cit.], the existence of a Weierstrass section \(e+V\) in \({\mathfrak {m}}\) was established by a general combinatorial construction. Notably, \(e \in {\mathscr {N}}\) and is a sum of root vectors with linearly independent roots. The Weierstrass section \(e+V\) 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 \({\mathscr {N}}^e\) of \({\mathscr {N}}\) 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 \(e_\textrm{VS}\) by adjoining root vectors. Then the linear span \(E_\textrm{VS}\) of these root vectors lies in \(\mathscr {N}^e\) and its closure is just \({\mathscr {N}}^e\). Yet, this same result shows that \({\mathscr {N}}^e\) 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 \({\mathscr {N}}^e\) 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 \({\mathscr {N}}\) 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.
期刊介绍:
The Journal of Algebraic Combinatorics provides a single forum for papers on algebraic combinatorics which, at present, are distributed throughout a number of journals. Within the last decade or so, algebraic combinatorics has evolved into a mature, established and identifiable area of mathematics. Research contributions in the field are increasingly seen to have substantial links with other areas of mathematics.
The journal publishes papers in which combinatorics and algebra interact in a significant and interesting fashion. This interaction might occur through the study of combinatorial structures using algebraic methods, or the application of combinatorial methods to algebraic problems.