A canonical basis for Garsia-Procesi modules
نویسنده
چکیده
We identify a subalgebra Ĥ + n of the extended affine Hecke algebra Ĥn of type A. The subalgebra Ĥ + n is a u-analogue of the monoid algebra of Sn n Z≥0 and inherits a canonical basis from that of Ĥn. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n, which we term positive affine tableaux (PAT). We then exhibit a cellular subquotient R1n of Ĥ + n that is a u-analogue of the ring of coinvariants C[y1, . . . , yn]/(e1, . . . , en) with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element π ∈ Ĥ + n corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that R1n has cellular quotients Rλ that are u-analogues of the Garsia-Procesi modules Rλ with left cells labeled by (a PAT version of) the λ-catabolizable tableaux. Résumé. On définit une sous-algèbre Ĥ + n de l’extension affine de l’algèbre de Hecke Ĥn de type A. La sousalgèbre Ĥ + n est u-analogue à l’algèbre monoı̈de de SnnZ≥0 et hérite d’une base canonique de Ĥn. On montre que ses cellules gauches sont naturellement classées par des tableaux remplis d’entiers naturels ayant chacun des restes différents modulo n, que l’on nomme Positive Affine Tableaux (PAT). On montre ensuite qu’un sous-quotient cellulaire R1n de Ĥ + n est une u-analogue de l’anneau des co-invariants C[y1, . . . , yn]/(e1, . . . , en) avec des cellules gauches classées PAT qui sont essentiellement des tableaux de Young standards avec des labels cochargés. Multiplier les éléments de la base canonique par un certain élément π ∈ Ĥ + n correspond à des rotations de mots, et par rapport aux cellules cela correspond à un cocyclage. Plus loin, on montre que R1n a pour quotients cellulaires Rλ qui sont uanalogues aux modules de Garsia-Procesi Rλ avec des cellules gauches définies par (une version PAT) des tableaux λ-catabolisable.
منابع مشابه
A Combinatorial Formula for the Hilbert Series of Bigraded Sn-Modules
We prove a combinatorial formula for the Hilbert series of the Garsia-Haiman bigraded Sn-modules as weighted sums over standard Young tableaux in the hook shape case. This method is based on the combinatorial formula of Haglund, Haiman and Loehr for the Macdonald polynomials and extends the result of A. Garsia and C. Procesi for the Hilbert series when q = 0. Moreover, we construct an associati...
متن کاملCombinatorial Formula for the Hilbert Series of bigraded Sn-modules
We introduce a combinatorial way of calculating the Hilbert series of bigraded Sn-modules as a weighted sum over standard Young tableaux in the hook shape case. This method is based on Macdonald formula for HallLittlewood polynomial and extends the result of A. Garsia and C. Procesi for the Hilbert series when q = 0. Moreover, we give the way of associating the fillings giving the monomial term...
متن کاملProject Summary: quantizing Schur functors
Geometric complexity theory (GCT) is an approach to P vs. NP and related problems in complexity theory using algebraic geometry and representation theory. A fundamental problem in representation theory, believed to be important for this approach, is the Kronecker problem, which asks for a positive combinatorial formula for the multiplicity gλμν of an irreducible representation Mν of the symmetr...
متن کاملA Hessenberg Generalization of the Garsia-Procesi Basis for the Cohomology Ring of Springer Varieties
The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this variety’s cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springer’s work more transparent and accessible by presenting the cohomology ring as a graded quotient of a poly...
متن کاملDescent Monomials, P-Partitions and Dense Garsia-Haiman Modules
A two-variable analogue of the descents monomials is defined and is shown to form a basis for the dense Garsia-Haiman modules. A two-variable generalization of a decomposition of a P-partition is shown to give the algorithm for the expansion into this descent basis. Some examples of dense Garsia-Haiman modules include the coinvariant rings associated with certain complex reflection groups.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010