arXiv subjects
Publications and source records attributed to Pooja Singla.
In this paper, we investigate abstract homomorphism from the special linear group over complete discrete valuation rings with finite residue field, such as the ring of p-adic integers, into the general linear group over the reals. We find the minimal dimension in which such a representation has infinite image. For positive characteristic rings, this minimum is infinity.
Let O be a complete discrete valuation ring with finite residue field k of odd characteristic. Let G be a general or special linear group or a unitary group defined over O and let $\mathfrak{g}$ denote its Lie algebra. For every positive integer l, let $K^l$ be the l-th principal congruence subgroup of G(O). A continuous irreducible representation of G(O) is called regular of level l if it is trivial on $K^{l+1}$ and its restriction to $K^l/K^{l+1} \simeq \mathfrak{g}(k)$ consists of characters with G(k)-stabiliser of minimal dimension. In this paper we construct the regular characters of G(O), compute their degrees and show that the latter satisfy Ennola duality. We give explicit uniform formulae for the regular part of the representation zeta functions of these groups.
Let F be a non-archimedean local field and let O be its ring of integers. We give a complete description of the irreducible constituents of the restriction of the unramified principal series representations of GL(3,F) to GL(3,O).
Let $R$ be a local principal ideal ring of length two, for example, the ring $R=\Z/p^2\Z$ with $p$ prime. In this paper we develop a theory of normal forms for similarity classes in the matrix rings $M_n(R)$ by interpreting them in terms of extensions of $R[t]$-modules. Using this theory, we describe the similarity classes in $M_n(R)$ for $n\leq 4$, along with their centralizers. Among these, we characterize those classes which are similar to their transposes. Non-self-transpose classes are shown to exist for all $n>3$. When $R$ has finite residue field of order $q$, we enumerate the similarity classes and the cardinalities of their centralizers as polynomials in $q$. Surprisingly, the polynomials representing the number of similarity classes in $M_n(R)$ turn out to have non-negative integer coefficients.
In this article we study the representations of general linear groups which arise from their action on flag spaces. These representations can be decomposed into irreducibles by proving that the associated Hecke algebra is cellular. We give a geometric interpretation of a cellular basis of such Hecke algebras which was introduced by Murphy in the case of finite fields. We apply these results to decompose representations which arise from the space of modules over principal ideal local rings of length two with a finite residue field.
We study the complex irreducible representations of special linear, symplectic, orthogonal and unitary groups over principal ideal local rings of length two. We construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions. The case for general linear groups has already been proved by author.
The centralizer algebra of a matrix consists of those matrices that commute with it. We investigate the basic representation-theoretic invariants of centralizer algebras, namely their radicals, projective indecomposable modules, injective indecomposable modules, simple modules and Cartan matrices. With the help of our Cartan matrix calculations we determine their global dimensions. Many of these algebras are of infinite global dimension.
We study the irreducible complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field. We construct a canonical correspondence between the irreducible representations of all such groups which preserves dimensions. For general linear groups of order three and four over these rings, we construct all the irreducible representations. We show that the the problem of constructing all the irreducible representations of all general linear groups over these rings is not easier than the problem of constructing all the irreducible representations of the general linear groups over principal ideal local rings of arbitrary length in the function field case.
Let $K$ be a field, $S$ a polynomial ring and $E$ an exterior algebra over $K$, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in $S$ and $E$ when passing to their generic initial ideals. First, we prove that if the graded Betti numbers $β_{ii+k}^S(S/I)=β_{ii+k}^S(S/\Gin(I))$ for some $i>1$ and $k \geq 0$, then $β_{qq+k}^S(S/I)= β_{qq+k}^S(S/\Gin(I))$ for all $q \geq i$, where $I\subset S$ is a graded ideal. Second, we show that if $β_{ii+k}^E(E/I)= β_{ii+k}^E(E/\Gin(I))$ for some $i>1$ and $k \geq 0$, then $β_{qq+k}^E(E/I)= β_{qq+k}^E(E/\Gin(I))$ for all $q \geq 1$, where $I\subset E$ is a graded ideal. In addition, it will be shown that the graded Betti numbers $β_{ii+k}^R(R/I)= β_{ii+k}^R(R/\Gin(I))$ for all $i \geq 1$ if and only if $I_{< k >}$ and $I_{< k+1 >}$ have a linear resolution. Here $I_{< d >}$ is the ideal generated by all homogeneous elements in $I$ of degree $d$, and $R$ can be either the polynomial ring or the exterior algebra.
A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley--Reisner ring has a linear resolution. It turns out that the Stanley--Reisner ring of the sphere which is the boundary complex of a linear ball satisfies the multiplicity conjecture. A class of shellable spheres arising naturally from commutative algebra whose Stanley--Reisner rings satisfy the multiplicity conjecture will be presented.
Let $I$ be a monomial ideal in a polynomial ring $A=K[x_1,...,x_n]$. We call a monomial ideal $J$ to be a minimal monomial reduction ideal of $I$ if there exists no proper monomial ideal $L \subset J$ such that $L$ is a reduction ideal of $I$. We prove that there exists a unique minimal monomial reduction ideal $J$ of $I$ and we show that the maximum degree of a monomial generator of $J$ determines the slope $p$ of the linear function $\reg(I^t)=pt+c$ for $t\gg 0$. We determine the structure of the reduced fiber ring $\mathcal{F}(J)_{\red}$ of $J$ and show that $\mathcal{F}(J)_{\red}$ is isomorphic to the inverse limit of an inverse system of semigroup rings determined by convex geometric properties of $J$.