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Symmetric functions, conjugacy classes and the flag variety
... In this section, we will investigate the closures of enhanced nilpotent orbits via establishing partial flags. Some strategies (see [1], or [12], [33], etc.) are exploited here. Keep the notations and assumptions as in the previous section. ...
... Enhanced version of De Concini-Procesi's theorem. De Concini-Procesi described the cohomology ring structure of Springer fibers in [12]. We present their result in the enhanced version. ...
... The statements directly follow from the classical results on Springer fibers (see [12], [19, §11.5], or [38]). ...
This is a sequel to \cite{osy} and \cite{sxy}. Associated with $G:=\GL_n$ and its rational representation $(\rho, M)$ over an algebraically closed filed $\bk$, we define an enhanced algebraic group $\uG:=G\ltimes_\rho M$ which is a product variety $\GL_n\times M$, endowed with an enhanced cross product. In this paper, we first show that the nilpotent cone $\ucaln:=\caln(\ugg)$ of the enhanced Lie algebra $\ugg:=\Lie(\uG)$ has finite nilpotent orbits under adjoint $\uG$-action if and only if up to tensors with one-dimensional modules, $M$ is isomorphic to one of the three kinds of modules: (i) a one-dimensional module, (ii) the natural module $\bk^n$, (iii) the linear dual of $\bk^n$ when $n>2$; and $M$ is an irreducible module of dimension not bigger than $3$ when $n=2$. We then investigate the geometry of enhanced nilpotent orbits when the finiteness occurs. Our focus is on the enhanced group $\uG=\GL(V)\ltimes_{\eta}V$ with the natural representation $(\eta, V)$ of $\GL(V)$, for which we give a precise classification of finite nilpotent orbits via a finite set $\scrpe$ of so-called enhanced partitions of $n=\dim V$, then give a precise description of the closures of enhanced nilpotent orbits via constructing so-called enhanced flag varieties. Finally, the $\uG$-equivariant intersection cohomology decomposition on the nilpotent cone of $\ugg$ along the closures of nilpotent orbits is established.
... De Concini and Procesi [9] and Tanisaki [42] found an explicit presentation for the graded ring H * (B λ ; Q) as a quotient R λ of the polynomial ring Q[x 1 , . . . , x n ]. ...
... As a special case, our construction gives a new compact geometric realization of the expression in the Delta Conjecture at t = 0. We also prove a version of the Springer correspondence for this family of varieties, showing that the top cohomology group has the S n -module structure of an induced Specht module. Finally, we generalize work of De Concini and Procesi [9] by introducing an ind-variety whose cohomology ring coincides with the coordinate ring of the scheme-theoretic intersection of an Eisenbud-Saltman rank variety with diagonal matrices. This is the full version of the extended abstract [16]. ...
... Finally, we generalize results of De Concini and Procesi [9]. Let gl n be the Lie algebra of n × n matrices over Q. ...
We introduce a family of varieties $Y_{n,\lambda,s}$, which we call the $\Delta$-Springer varieties, that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring $H^*(Y_{n,\lambda,s})$ and show that there is a symmetric group action on this ring generalizing the Springer action on the cohomology of a Springer fiber. In particular, the top cohomology groups are induced Specht modules. The $\lambda=(1^k)$ case of this construction gives a compact geometric realization for the expression in the Delta Conjecture at $t=0$. Finally, we generalize results of De Concini and Procesi on the scheme of diagonal nilpotent matrices by constructing an ind-variety $Y_{n,\lambda}$ whose cohomology ring is isomorphic to the coordinate ring of the scheme-theoretic intersection of an Eisenbud-Saltman rank variety and diagonal matrices.
... There is also an algebraic approach to the Springer representation for GL n (C), as we now explain. Motivated by a conjecture of Kraft [25], De Concini and Procesi [8] gave a presentation for the cohomology of a type A Springer fiber as the quotient of a polynomial ring. Furthermore, this identification is S n -equivariant so Springer's representation can also be constructed as the symmetric group action on the quotient of a polynomial ring. ...
... Their work gives a linear algebraic proof that this character is closely connected to the so-called q-Kostka polynomials. As part of their analysis, Garsia and Procesi study a monomial basis for the cohomology ring, originally defined by De Concini and Procesi in [8], with many amenable combinatorial and 482 MARTHA PRECUP AND EDWARD RICHMOND inductive properties. We refer to the collection of these monomials as the Springer monomial basis. ...
... We now recall the monomial basis of A 0 H * (B α ) defined by De Concini and Procesi in [8] and further analyzed by Garsia and Procesi in [19]. ...
Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for G L n ( C ) GL_n(\mathbb {C}) using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equivariant and ordinary cohomology rings of Springer fibers. As an application, we identify an explicit collection of (equivariant) Schubert classes whose images in the (equivariant) cohomology ring of a given Springer fiber form a basis.
... The space B μ is a fiber of the Springer resolution of the unipotent subvariety of GL n , and its cohomology ring comes with a graded S n -module structure whose top degree component is precisely the irreducible representation V μ [21]. The work of [3] and [22] shows that R μ is isomorphic to the cohomology ring of the Springer fiber B μ , both as a graded ring and as a graded S n -module. ...
... where t is the Cartan subalgebra of diagonal matrices in gl n . This is a strict generalization of the analogous result for R μ and O μ , which was an essential step in de Concini and Procesi's work [3] on the connections to the Springer fibers. ...
... We finally label w (3) to obtain ...
The classical coinvariant ring \(R_n\) is defined as the quotient of a polynomial ring in n variables by the positive-degree \(S_n\)-invariants. It has a known basis that respects the decomposition of \(R_n\) into irreducible \(S_n\)-modules, consisting of the higher Specht polynomials due to Ariki, Terasoma, and Yamada (Hiroshima Math J 27(1):177–188, 1997). We provide an extension of the higher Specht basis to the generalized coinvariant rings \(R_{n,k}\) introduced in Haglund et al. (Adv Math 329:851–915, 2018). We also give a conjectured higher Specht basis for the Garsia–Procesi modules \(R_\mu \), and we provide a proof of the conjecture in the case of two-row partition shapes \(\mu \). We then combine these results to give a higher Specht basis for an infinite subfamily of the modules \(R_{n,k,\mu }\) recently defined by Griffin (Trans Amer Math Soc, to appear, 2020), which are a common generalization of \(R_{n,k}\) and \(R_{\mu }\).
... De Concini and Procesi [6] gave a description of H * (F λ ; C) as the coordinate ring of an (unreduced) variety over C which we now describe. Let λ ∨ denote the partition dual to λ. ...
... The coordinate ring C[t C ∩ O λ ∨ ] of the (non-reduced) scheme t C ∩ O λ ∨ (scheme theoretic intersection) where t = Lie(T n C ) ⊂ gl(n, C) = M n (C) and O λ ⊂ M n (C) denotes the closure of the orbit of J λ ∨ under the adjoint action of GL(n, C). De Concini and Procesi showed that H * (F λ ; C) is isomorphic to the algebra C[t ∩ O λ ∨ ] (see [6]). ...
The aim of this paper is to describe the topological equivariant $K$-ring, in terms of generators and relations, of a Springer variety $\mathcal{F}_{\lambda}$ of type $A$ associated to a nilpotent operator having Jordan canonical form whose block sizes form a weakly decreasing sequence $\lambda=(\lambda_1,\ldots, \lambda_l)$. This parallels the description of the equivariant cohomology ring of $\mathcal{F}_{\lambda}$ due to Abe and Horiguchi and generalizes the description of ordinary topological $K$-ring of $\mathcal{F}_{\lambda}$ due to Sankaran and Uma \cite{su}.
... De Concini and Procesi [5] gave a description of H * (F λ ; C) as the coordinate ring of an (unreduced) variety over C which we now describe. Let λ ∨ denote the partition dual to λ. ...
... For an arbitrary partition λ = λ 1 ≥ · · · ≥ λ l of n, the Springer variety F λ is naturally imbedded in F. The cohomology ring of the Springer variety F λ has been described by Tanisaki [16] in terms of generators and relations, in a way that generalizes the above description of H * (F; Z). It turns out that, although Tanisaki considered cohomology with complex coefficients, his description recalled below, is valid even when the coefficient ring is the integers (see [5]). We need the following notation. ...
The aim of this paper is to describe the topological $K$-ring, in terms of generators and relations, of a Springer variety $\mathcal{F}_{\lambda}$ of type $A$ associated to a nilpotent operator having Jordan canonical form whose block sizes forms a weakly decreasing sequence $\lambda=(\lambda_1,\ldots, \lambda_l)$. Our description parallels the description of the integral cohomology ring of $\mathcal{F}_{\lambda}$ due to Tanisaki and also the equivariant analogue due to Abe and Horiguchi.
... In this paper, we consider the Hikita conjecture in the case (2). In this case, while the structure of the cohomology rings H * (M r,n , C) and H * (Hilb n ( C 2 /(Z/rZ)), C) are well known, the structure of the coordinate rings of the fixed are not yet clean. ...
... In this case, while the structure of the cohomology rings H * (M r,n , C) and H * (Hilb n ( C 2 /(Z/rZ)), C) are well known, the structure of the coordinate rings of the fixed are not yet clean. Hence we compare these objects as graded vector spaces which is the case (2). Our purpose is to prove the following theorem. ...
If two conical symplectic resolutions $X\to X_0$ and $X^!\to X_0^!$ are symplectic dual, the cohomology ring $H^*(X)$ and the coordinate ring of $\mathbb{C}^*$-fixed points in $X_0^!$ are expected to be isomorphic as graded algebras. This statement is called Hikita conjecture and it is known that the conjecture holds for some cases. In this paper, we deal with the cohomology of framed moduli spaces over the projective plane and the coordinate ring of $\mathbb{C}^*$- fixed points of $\mathbb{C}^{2n}/((\mathbb{Z}/r\mathbb{Z})\wr S_n) $ and show that these are isomorphic as graded vector spaces.
... The representation theoretical setting of Hall-Littlewood polynomials can be found through the work of Steinberg [43], Hotta and Springer [32], Kraft [34], and De Concini and Procesi [12]. These proofs rely on algebraic geometry. ...
The characters of the (total) Springer representations are identified with the Green functions by Kazhdan (1977), and the latter are identified with Hall-Littlewood’s Q-functions by Green (1955). In this paper, we present a purely algebraic proof that the (total) Springer representations of GL(n) are Ext-orthogonal to each other, and show that it is compatible with the natural categorification of the ring of symmetric functions.
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].
