General Linear Group Research Articles

On enhanced general linear groups: nilpotent orbits and support variety for Weyl module

<abstract><p>Associated with a reductive algebraic group $ G $ and its rational representation $ (\rho, M) $ over an algebraically closed filed $ {\bf{k}} $, the authors define the enhanced reductive algebraic group $ {\underline{G}}: = G\ltimes_\rho M $, which is a product variety $ G\times M $ and endowed with an enhanced cross product in <sup>[<xref ref-type="bibr" rid="b5">5</xref>]</sup>. If $ {\underline{G}} = GL(V)\ltimes_{\eta} V $ with the natural representation $ (\eta, V) $ of $ {\text{GL}}(V) $, it is called an enhanced general linear algebraic group. And the authors give a precise classification of finite nilpotent orbits via a finite set of so-called enhanced partitions of $ n = \dim V $ for the enhanced group $ {\underline{G}} = {\text{GL}}(V)\ltimes_{\eta}V $ in [<xref ref-type="bibr" rid="b6">6</xref>, Theorem 3.5]. We will give another way to prove this classification theorem in this paper. Then we focus on the support variety of the Weyl module for $ {\underline{G}} = {\text{GL}}(V)\ltimes_{\eta}V $ in characteristic $ p $, and obtain that it coinsides with the closure of an enhanced nilpotent orbit under some mild condition.</p></abstract>

Equivalence of 2-tensors and 3-tensors under Local Unitary and General Linear Groups

Equivalence of 2-tensors and 3-tensors under Local Unitary and General Linear Groups

When are the natural embeddings of classical invariant rings pure?

Abstract Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$ -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

Totally Symmetric Sets in the General Linear Group

Totally Symmetric Sets in the General Linear Group

Левоинвариантная парасасакиева структура на групповой модели вещественного расширения плоскости де-Ситтера

In this paper, a group model for a real extension of the de Sitter plane is pro-posed. This group contains a group of special matrices, which is a subgroup of the general linear group. It is established that there exists a left-invariant contact metric structure on this group, which is normal and, therefore, para-Sasakian. The basis vector fields of the Lie algebra of infinitesimal automorphisms are found. The Lie group of automorphisms has the maximum dimension and, in addition to the Levi-Civita connection, it also retains a contact metric connection with skew-symmetric torsion. In this connection, all structural tensors of the para-Sasakian structure, as well as the torsion and curvature tensors, are covariantly constant. Using a nonholonomic field of orthonormal frames adapted to the contact distribution, an orthogonal projection of the Levi-Civita connection onto the contact distribution is found, which is a truncated connection. Passing to natural coordinates, differential equations of geodesics of the truncated connection and Levi-Civita connection are found. Thus, the Levi-Civita contact geodesic connections coincide with the truncated connection geodesics. This means that through each point in each contact direction there is a unique Levi-Civita geodesic connection tangent to the contact distribution. The Levi-Civita connection, like the contact metric connection, is consistent with the contact distribution.

Appearance of the Kashiwara–Saito singularity in the representation theory of p-adic GL(16)

In 1993 David Vogan proposed a basis for the vector space of stable distributions on $p$-adic groups using the microlocal geometry of moduli spaces of Langlands parameters. In the case of general linear groups, distribution characters of irreducible admissible representations, taken up to equivalence, form a basis for the vector space of stable distributions. In this paper we show that these two bases, one putative, cannot be equal. Specifically, we use the Kashiwara-Saito singularity to find a non-Arthur type irreducible admissible representation of $p$-adic $\mathop{GL}_{16}$ whose ABV-packet, as defined in earlier work, contains exactly one other representation; remarkably, this other admissible representation is of Arthur type. In the course of this study we strengthen the main result concerning the Kashiwara-Saito singularity. The irreducible admissible representations in this paper illustrate a fact we found surprising: for general linear groups, while all A-packets are singletons, some ABV-packets are not.

Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group

Consider the random walk Gn:=gn…g1, n⩾1, where (gn)n⩾1 is a sequence of independent and identically distributed random elements with law μ on the general linear group GL(V) with V=Rd. Under suitable conditions on μ, we establish Cramér type moderate deviation expansions and local limit theorems with moderate deviations for the coefficients 〈f,Gnv〉, where v∈V and f∈V∗. Our approach is based on the Hölder regularity of the invariant measure of the Markov chain Gn⋅x=RGnv on the projective space of V with the starting point x=Rv, under the changed measure.

Overgroups of exterior powers of an elementary group. levels

ABSTRACT We prove a first part of the standard description of groups H lying between an exterior power of an elementary group m E n ⁡ ( R ) and a general linear group GL ( n m ) ⁡ ( R ) for a commutative ring R , 2 ∈ R ∗ and n ⩾ 3 m . The description uses the classical notion of a level: for every group H we find a unique ideal A of the ground ring R, which describes H.

Counting formulas for certain p-subgroups of GL_n(F_p)

Let p be a prime number and F_p a finite field of order p. Let GL_n(F_p) denote the general linear group and let U_n denote the unitriangular group of n x n upper triangular matrices with ones on the diagonal, over the finite field F_p. This is a finite group of order and a Sylow p-subgroup of GL_n(F_p}. In this work, we characterize some p-subgroups of GL_n(F_p) with respect to a given property.

Spin 1/2 one‐ and two‐particle systems in physical space without eigen‐algebra or tensor product

Under the spin–position decoupling approximation, a vector with a phase in 3D orientation space endowed with geometric algebra substitutes the vector–matrix spin model built on the Pauli spin operator. The standard quantum operator‐state spin formalism is replaced with vectors transforming by proper and improper rotations in the same 3D space—isomorphic to the space of Pauli matrices. In the single‐spin case, the novel spin 1/2 representation (1) is Hermitian, (2) shows handedness, (3) yields all the standard results and its modulus equals the total spin angular momentum , (4) formalizes irreversibility in measurement, and (5) permits adaptive imbedding of the 2D spin space in 3D. Maximally entangled spin pairs (1) are in phase and have opposite handedness, (2) relate by one of the four basic improper rotations in 3D: plane reflections (triplets) and inversion (singlet), (3) yield the standard total angular momentum, and (4) all standard expectation values for bipartite and partial observations follow. Depending on whether proper and improper rotors act one—or two—sided, the formalism appears in two complementary forms, the “spinor” or the “vector” form, respectively. The proposed scheme provides a clear geometric picture of spin correlations and transformations entirely in the 3D physical orientation space.

Simple modules for quiver Hecke algebras and the Robinson–Schensted–Knuth correspondence

We formalize some known categorical equivalences to give a rigorous treatment of smooth representations of p-adic general linear groups, as ungraded modules over quiver Hecke algebras of type A. Graded variants of RSK-standard modules are constructed for quiver Hecke algebras. Exporting recent results from the p-adic setting, we describe an effective method for construction and classification of all simple modules as quotients of modules induced from maximal homogenous data. It is established that the products involved in the RSK construction fit the Kashiwara-Kim notion of normal sequences of real modules. We deduce that RSK-standard modules have simple heads, devise a formula for the shift of grading between RSK-standard and simple self-dual modules, and establish properties of their decomposition matrix, thus confirming expectations for p-adic groups raised in a work of the author with Lapid. Subsequent work will exhibit how the presently introduced RSK construction generalizes the much-studied Specht construction, when inflated from cyclotomic quotient algebras.

Kohnert's rule for flagged Schur modules

Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure characters, flagged skew Schur polynomials, and Schubert polynomials. In this paper, we prove the characters of flagged Schur modules can be computed using a simple combinatorial algorithm due to Kohnert if and only if the indexing diagram is northwest. This gives a new proof that characters of flagged Schur modules are nonnegative sums of Demazure characters and gives a representation theoretic interpretation for Kohnert polynomials.

A zero-one law for invariant measures and a local limit theorem for coefficients of random walks on the general linear group

Nous prouvons une loi zéro-un pour la mesure stationnaire pour des ensembles algébriques en généralisant les résultats de Furstenberg (Proc. Sympos. Pure Math. 26 (1973) 193–229) et Guivarc’h et Le Page (Ann. Inst. Henri Poincaré Probab. Stat. 52(2) (2016) 503–574). Comme application, nous établissons un théorème local limite pour les coefficients de marches aléatoires sur le groupe linéaire général.

Modeling of Spin Crossover in Linear Trinuclear Complexes: Effects of External Pressure.

In the present study, a model is suggested to explain the course of spin transformation in crystals containing linear trinuclear iron(II) clusters as structural units. The energy spectrum of an isolated trimer is described with due allowance of the effects of the cubic crystal field formed by the nearest surrounding of each FeII ion as well as by consideration of intracluster interactions in the nearest neighbor approximation. The intercluster cooperative interaction promoting the spin transition is assumed to arise from the coupling of molecular modes with phonons, that is, a feature characteristic for molecular crystals containing as a structural element single spin-crossover ions or complexes formed by these ions. It is demonstrated that in dependence on the relative strengths of intra- and intercluster interactions, the trinuclear systems can exhibit a variety of different types of spin transitions and, namely, gradual, step-like transitions and those characterized by a hysteresis loop. The effects of external pressure on the spin transitions in crystals of trinuclear clusters are explored taking into account that pressure affects both the energy gap between the high-spin and low-spin states and the characteristic parameters of cooperative interaction. In the framework of the suggested model, the spin transformation in the Fe3(bntrz)6(tcnset)6 compound is comprehensively explained. The model also allows a qualitative and quantitative reproduction of the temperature and course of the spin transition in this compound under applied pressure.

Geometric Howe dualities of finite type

We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp. type B/C), the Howe duality between a pair of quantum general linear groups (resp. a pair of ıquantum groups of type AIII/IV) is established. The Howe duality for quantum general linear groups has been provided via quantum coordinate algebras in [33]. We also generalize this algebraic approach to ıquantum groups of type AIII/IV, and prove that the quantum Howe duality derived from partial flag varieties coincides with the one constructed by quantum coordinate (co)algebras. Moreover, the explicit multiplicity-free decompositions for these Howe dualities are obtained.

Chromatic (co)homology of finite general linear groups

We study the Morava $E$-theory (at a prime $p$) of $BGL_d(F)$, where $F$ is a finite field with $|F|=1\pmod{p}$. Taking all $d$ together, we obtain a structure with two products $\times$ and $\bullet$. We prove that it is a polynomial ring under $\times$, and that the module of $\times$-indecomposables inherits a $\bullet$-product, and we describe the structure of the resulting ring. In the process, we prove many auxiliary structural results.

Some Erdös-Ko-Rado results for linear and affine groups of degree two

In this paper, we show that both the general linear group GL(2,q) and the special linear group SL(2,q) have both the EKR property and the EKR-module property. This is done using an algebraic method; a weighted adjacency matrix for the derangement graph for the group is found and Hoffman’s ratio bound is applied to this matrix. We also consider the group AGL(2,q) and the 2-intersecting sets in PGL(2,q).

A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

Let A be the classical, singly-graded Steenrod algebra over the prime order field F2 and let P⊗h:=F2[t1,…,th] denote the polynomial algebra on h generators, each of degree 1, viewed as a module over A. Write GLh for the usual general linear group of rank h over F2. As well known, the (mod 2) cohomology groups of the Steenrod algebra, ExtAh,h+⁎(F2,F2) is still largely mysterious for all homological degrees h≥6. The h-th algebraic transferTrhA:(F2⊗GLhAnnA‾[P⊗h]⁎)n⟶ExtAh,h+n(F2,F2), defined by William Singer [28], is a helpful tool to describe that Ext groups. Singer conjectured that this transfer is a monomorphism, but it remains open for any h≥5. There is currently no information on the conjecture for h=6. In this paper, we verify Singer's conjecture for homological degree h=6 and the internal degree of the general form ns:=6(2s−1)+10⋅2s. This result is important, since it tells us that the non-zero elements h22g1=h4Ph2∈ExtA6,6+n1(F2,F2), and D2∈ExtA6,6+n2(F2,F2) are not in the image of the sixth algebraic transfer. This Note is a continuation of our previous one [22], which will refer to as Part I. Our approach is based on explicitly solving the hit problem for the Steenrod algebra in the case h=6 and degree ns. This extends a result in Mothebe, Kaelo and Ramatebele [13]. At the same time, we also use this obtained result to establish the dimension result for the space of the unhit elements, F2⊗AP⊗7 in a certain general degree.

Tangent Spaces of Orbit Closures for Representations of Dynkin Quivers of Type mathbb {D}

Let Bbbk be an algebraically closed field, Q a finite quiver, and denote by textup {rep}_{Q}^{mathbf {d}} the affine Bbbk -scheme of representations of Q with a fixed dimension vector d. Given a representation M of Q with dimension vector d, the set {mathcal {O}}_{M} of points in Bbbk isomorphic as representations to M is an orbit under an action on textup {rep}^{mathbf {d}}_{Q}Bbbk of a product of general linear groups. The orbit {mathcal {O}}_{M} and its Zariski closure overline {mathcal {O}}_{M}, considered as reduced subschemes of textup {rep}_{Q}^{{mathbf {d}}}, are contained in an affine scheme {mathcal {C}}_{M} defined by suitable rank conditions associated to M. For all Dynkin and extended Dynkin quivers, the sets of points of overline {{mathcal {O}}}_{M} and {mathcal {C}}_{M} coincide, or equivalently, overline {{mathcal {O}}}_{M} is the reduced scheme associated to {mathcal {C}}_{M}. Moreover, overline {mathcal {O}}_{M}={mathcal {C}}_{M} provided Q is a Dynkin quiver of type {mathbb {A}}, and this equality is a conjecture for the remaining Dynkin quivers (of type mathbb {D} and {mathbb {E}}). Let Q be a Dynkin quiver of type mathbb {D} and M a finite dimensional representation of Q. We show that the equality T_{N}overline {mathcal {O}}_{M}=T_{N}{mathcal {C}}_{M} of Zariski tangent spaces holds for any closed point N of overline {mathcal {O}}_{M}. As a consequence, we describe the tangent spaces to overline {mathcal {O}}_{M} in representation theoretic terms.

The affirmative answer to Singer's conjecture on the algebraic transfer of rank four

During the last decades, the structure of mod-2 cohomology of the Steenrod ring $\mathscr {A}$ became a major subject in Algebraic topology. One of the most direct attempt in studying this cohomology by means of modular representations of the general linear groups was the surprising work [Math. Z.202 (1989), 493–523] by William Singer, which introduced a homomorphism, the so-called algebraic transfer, mapping from the coinvariants of certain representation of the general linear group to mod-2 cohomology group of the ring $\mathscr A.$ He conjectured that this transfer is a monomorphism. In this work, we prove Singer's conjecture for homological degree $4.$