An official website of the United States government
Abstract Each connected graded, graded-commutative algebraAof finite type over a field$$\Bbbk $$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the(higher) Koszul modulesofA. In this note, we investigate the geometry of the support loci of these modules, called theresonance schemesof the algebra. When$$A=\Bbbk \langle \Delta \rangle $$ is the exterior Stanley–Reisner algebra associated to a finite simplicial complex$$\Delta $$ , we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.
more »
« less
A Proof of the Extended Delta Conjecture
Abstract We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $$\Delta _{h_l}\Delta ' _{e_k} e_{n}$$ , where $$\Delta ' _{e_k}$$ and $$\Delta _{h_l}$$ are Macdonald eigenoperators and $$e_n$$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $$\operatorname {\mathrm {GL}}_m$$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.
more »
« less
- PAR ID:
- 10410650
- Date Published:
- Journal Name:
- Forum of Mathematics, Pi
- Volume:
- 11
- ISSN:
- 2050-5086
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We prove that $$\omega \Delta ^{\prime}_{e_{k}}e_{n}|_{t=0}$$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all $$\Delta $$-Springer modules. We use this to give an explicit Schur expansion in terms of the Lascoux-Schützenberger cocharge statistic on a new combinatorial object that we call a battery-powered tableau. Our proof is geometric, and shows that the $$\Delta $$-Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $$t$$ and $$t^{2}$$ coefficients of $$\omega \Delta ^{\prime}_{e_{k}}e_{n}$$.more » « less
-
Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.more » « less
-
Abstract We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the$$\mathscr {J}$$ -classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra.more » « less
-
Abstract We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath MacdonaldP-polynomials; the eigenvalues are written in terms of elementary symmetric polynomials of arbitrary degree. Our operators arise from integral formulas for the action of the horizontal Heisenberg subalgebra in the vertex representation of the corresponding quantum toroidal algebra.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fmp.2023.3
