ArticlePDF Available
A preview of the PDF is not available
... (See [2,7,8]. See also [1,9] for algebraic construction.) Let µ be an l-partition, where an l-partition means a partition whose multiplicities are divisible by l. ...
... , l n ) be n-tuples of positive integers, m the sum h m h , l the least common multiple of { l i } and ζ k the primitive kth root of unity. We call an n-tuple µ = (µ (1) ...
... The set of numberings whose components are arranged in ascending order in each row is a complete set of representatives for row-equivalence classes. A row-equivalence class of a numbering T on an l-partition µ is an n-tuple ({T (1) ...
Preprint
We consider certain modules of the symmetric groups whose basis elements are called tabloids. Some of these modules are isomorphic to subspaces of the cohomology rings of subvarieties of flag varieties as modules of the symmetric groups. We give a combinatorial description for some weighted sums of their characters, i.e., we introduce combinatorial objects called (\rho,\Ll)-tabloids and rewrite weighted sums of characters as the numbers of these combinatorial objects. We also consider the meaning of these combinatorial objects, i.e., we construct a correspondence between (\rho,\Ll)-tabloids and tabloids whose images are eigenvectors of the action of an element of cycle type ρ\rho in quotient modules.
... Since changing the orientations of the merges does not change the result of the functor, we will ignore them in our discussion. The cohomology ring of B λ can be computed by quotienting the polynomial ring in m variables by the ideal of partially symmetric functions (see [13] for more details). Write (n, n) for the partition λ = (n, n) of 2n. ...
... Theorem 5. 13. For all C ∈ C n there exists a map τ C : N 2 (G 2 × Z) → Z/4Z such that the twisted algebra (OH n C ⊗ Z Z[i]) τ C is associative. ...
Preprint
We construct an odd version of Khovanov's arc algebra HnH^n. Extending the center to elements that anticommute, we get a subalgebra that is isomorphic to the oddification of the cohomology of the (n,n)-Springer varieties. We also prove that the odd arc algebra can be twisted into an associative algebra.
... In [13], the author and Rhoades found higher Specht bases for both the modules R n,k (the Haglund-Rhoades-Shimozono modules defined in the context of the t = 0 specialization of the Delta conjecture [15]) and R µ (the Garsia-Procesi modules, which are the cohomology rings of the fibers of the Springer resolution [9,11,25]). They proved their construction was a basis in the former case and conjectured for the latter, proving it for µ having two parts. ...
... It has a combinatorial formula involving a beautiful statistic known as charge [LS78]: where the sum is taken over all semistandard Young tableaux of shape λ with weight µ. The Kostka-Foulkes polynomial has its origins in geometry and representation theory, with significant connections to flag varieties and Springer fibers [HS77,DCP81,Tan82,GP92]. Due to its ubiquity and rich structure, the study of the Kostka-Foulkes polynomial has been an active area of research for decades. ...
Preprint
Lusztig q-weight multiplicities extend the Kostka-Foulkes polynomials to a broader range of Lie types. In this work, we investigate these multiplicities through the framework of Kirillov-Reshetikhin crystals. Specifically, for type C with dominant weights and type B with dominant spin weights, we present a combinatorial formula for Lusztig q-weight multiplicities in terms of energy functions of Kirillov-Reshetikhin crystals, generalizing the charge statistic on semistandard Young tableaux for type A. Additionally, we introduce level-restricted q-weight multiplicities for nonexceptional types, and prove positivity by providing their combinatorial formulas.
... Since the odd-dimensional Betti numbers of Springer varieties are zero (cf. [3]), the same must be true for those of Spaltenstein varieties. In particular, it follows that H 1 (M ) = 0. ...
Preprint
For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conjecture them to be equal to those of Khovanov and Rozansky after a collapsation of the bigrading. Our work is a generalization of that of Seidel and Smith, who treated the case n=2.
... Our result gives a good set of generators for the ideal I 0 L in the case where L is a partition µ or a partition with a hole µ/ij. The case L = µ was studied extensively in [7,8,11,15]. Our description is dual to Tanisaki's [7,15]. ...
Preprint
A lattice diagram is a finite set L={(p1,q1),...,(pn,qn)}L=\{(p_1,q_1),... ,(p_n,q_n)\} of lattice cells in the positive quadrant. The corresponding lattice diagram determinant is \Delta_L(\X;\Y)=\det \| x_i^{p_j}y_i^{q_j} \|. The space MLM_L is the space spanned by all partial derivatives of \Delta_L(\X;\Y). We denote by ML0M_L^0 the Y-free component of MLM_L. For μ\mu a partition of n+1, we denote by μ/ij\mu/ij the diagram obtained by removing the cell (i,j) from the Ferrers diagram of μ\mu. Using homogeneous partially symmetric polynomials, we give here a dual description of the vanishing ideal of the space Mμ0M_\mu^0 and we give the first known description of the vanishing ideal of Mμ/ij0M_{\mu/ij}^0.
Preprint
We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form h=(h(1),n,n)h=(h(1),n\dots,n) in Lie type An1A_{n-1}. The main result of this paper gives an explicit presentation of the cohomology rings in terms of generators and their relations. Our presentation naturally specializes to Borel's presentation of the cohomology ring of the flag variety and it is compatible with the representation of the symmetric group Sn\mathfrak{S}_n on the cohomology constructed by J. Tymoczko. As a corollary, we also give an explicit presentation of the Sn\mathfrak{S}_n-invariant subring of the cohomology ring.
Preprint
Local models are schemes defined in linear algebra terms that describe the 'etale local structure of integral models for Shimura varieties and other moduli spaces. We point out that the flatness conjecture of Rapoport-Zink on local models fails in the presence of ramification and that in that case one has to modify their definition. We study in detail certain modifications of the local models for G=R_{E/F}GL(n), with E/F a totally ramified extension, and for a maximal parahoric level subgroup. The special fibers of these models are subschemes of the affine Grassmannian. We show that the new local models are smoothly equivalent to "rank varieties" of matrices, are flat, normal, with rational singularities and that their special fibers contain the expected Schubert strata. A corollary is that Schubert varieties in the affine Grassmannian are smoothly equivalent to nilpotent orbit closures and are normal with rational singularities, even in positive characteristics. We give some applications to the calculation of sheaves of nearby cycles and describe a relation with geometric convolution. Finally, in the general EL case, we replace the flatness conjecture of Rapoport-Zink with a conjecture about the modified local models.
Preprint
In this article we calculate the signature character of certain Hermitian representations of GLN(F)GL_N(F) for a p-adic field F. We further give a conjectural description for the signature character of unramified representations in terms of Kostka numbers.
Preprint
This note is motivated by the problem of understanding Springer's remarkable action of the Weyl group W=NG(T)/TW=N_G(T)/T of a semi-simple complex linear algebraic group G, with maximal torus T, on the cohomology algebra of an arbitrary Springer variety in the flag variety of G from the viewpoint of torus actions. Continuing the work [CK] which gave a sufficient condition for a group W\mathcal{W} acting on the fixed point set of an algebraic torus action (S,X) on a complex projective variety X to lift to a representation of W\mathcal{W} on the cohomology algebra H(X)H^*(X) (over C\mathbb{C}), we describe when the representation on H(X)H^*(X) is equivalent to the representation of W\mathcal{W} on the cohomology H(XS)H^*(X^S) of the fixed point set. As a consequence of this theorem, we give a simple proof in type A of the Alvis-Lusztig-Treumann Theorem, which describes Springer's representation of W for Springer varieties corresponding to nilpotents in a Levi subalgebra of Lie(G). In the final two sections, we describe the local structure of the moment graph M(X)\mathfrak{M}(X) of a special torus action (S,X), and we also show that if a finite group W\mathcal{W} acts on the moment graph of X, then W\mathcal{W} induces pair of actions on H(X)H^*(X), namely the left and right or dot and star actions of Knutson [Knu] and Tymoczko [Tym] respectively. In particular, W acts on the moment (or Bruhat) graph M(G/P)\mathfrak{M}(G/P) of (T,G/P) for any parabolic P in G containing T, and the right action of W on H(G/P)H^*(G/P) is an induced representation. Furthermore, we show the left action of W on H(G/P)H^*(G/P) is trivial.
Chapter
No group is of greater importance than the symmetric group. After all, any group can be embedded as a subgroup of a symmetric group. In this chapter, we construct the irreducible representations of the symmetric group S n . The character theory of the symmetric group is a rich and important theory filled with important connections to combinatorics. One can find whole books dedicated to this subject, cf. [16, 17, 19, 12]. Moreover, there are important applications to such diverse areas as voting and card shuffling [7, 8, 3].