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| Formato: | Preprint |
| Publicado: |
2003
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| Acceso en línea: | https://arxiv.org/abs/math/0310121 |
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| _version_ | 1866917487254175744 |
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| author | Reading, Nathan |
| author_facet | Reading, Nathan |
| contents | We study flag enumeration in intervals in the Bruhat order on a Coxeter group by means of a structural recursion on intervals in the Bruhat order. The recursion gives the isomorphism type of a Bruhat interval in terms of smaller intervals, using basic geometric operations which preserve PL sphericity and have a simple effect on the cd-index. This leads to a new proof that Bruhat intervals are PL spheres as well a recursive formula for the cd-index of a Bruhat interval. This recursive formula is used to prove that the cd-indices of Bruhat intervals span the space of cd-polynomials.
The structural recursion leads to a conjecture that Bruhat spheres are "smaller" than polytopes. More precisely, we conjecture that if one fixes the lengths of x and y, then the cd-index of a certain dual stacked polytope is a coefficientwise upper bound on the cd-indices of Bruhat intervals [x,y]. We show that this upper bound would be tight by constructing Bruhat intervals which are the face lattices of these dual stacked polytopes. As a weakening of a special case of the conjecture, we show that the flag h-vectors of lower Bruhat intervals are bounded above by the flag h-vectors of Boolean algebras (i.e. simplices). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0310121 |
| institution | arXiv |
| publishDate | 2003 |
| record_format | arxiv |
| spellingShingle | The cd-index of Bruhat intervals Reading, Nathan Combinatorics 20F55, 06A07 We study flag enumeration in intervals in the Bruhat order on a Coxeter group by means of a structural recursion on intervals in the Bruhat order. The recursion gives the isomorphism type of a Bruhat interval in terms of smaller intervals, using basic geometric operations which preserve PL sphericity and have a simple effect on the cd-index. This leads to a new proof that Bruhat intervals are PL spheres as well a recursive formula for the cd-index of a Bruhat interval. This recursive formula is used to prove that the cd-indices of Bruhat intervals span the space of cd-polynomials. The structural recursion leads to a conjecture that Bruhat spheres are "smaller" than polytopes. More precisely, we conjecture that if one fixes the lengths of x and y, then the cd-index of a certain dual stacked polytope is a coefficientwise upper bound on the cd-indices of Bruhat intervals [x,y]. We show that this upper bound would be tight by constructing Bruhat intervals which are the face lattices of these dual stacked polytopes. As a weakening of a special case of the conjecture, we show that the flag h-vectors of lower Bruhat intervals are bounded above by the flag h-vectors of Boolean algebras (i.e. simplices). |
| title | The cd-index of Bruhat intervals |
| topic | Combinatorics 20F55, 06A07 |
| url | https://arxiv.org/abs/math/0310121 |