On natural representations of the symplectic group

2013

For k = 1, 2, ..., n−1 let V k = V (λ k) be the Weyl module for the special orthogonal group G = SO(2n + 1, F) with respect to the k-th fundamental dominant weight λ k of the root system of type Bn and put Vn = V (2λn). It is well known that all of these modules are irreducible when char(F) = 2 while when char(F) = 2 they admit many proper submodules. In this paper, assuming that char(F) = 2, we prove that V k admits a chain of submodules V k = M k ⊃ M k−1 ⊃ ... ⊃ M1 ⊃ M0 ⊃ M−1 = 0 where Mi ∼ = Vi for 1, ..., k−1 and M0 is the trivial 1-dimensional module. We also show that for i = 1, 2, ..., k the quotient Mi/Mi−2 is isomorphic to the so called i-th Grassmann module for G. Resting on this fact we can give a geometric description of Mi−1/Mi−2 as a submodule of the i-th Grassmann module. When F is perfect G ∼ = Sp(2n, F) and Mi/Mi−1 is isomorphic to the Weyl module for Sp(2n, F) relative to the i-th fundamental dominant weight of the root system of type Cn. All irreducible sections of the latter modules are known. Thus, when F is perfect, all irreducible sections of V k are known as well.

Journal of Algebra, 2002

Journal of Algebra, 1999

2021

The irreducible representations ϕ_n^1 and ϕ_n^2 of the symplectic group G_n=Sp_2n(P) over an algebraically closednfield P of characteristic p>2 with highest weights ω_n-1+p-3/2ω_n and p-1/2ω_n, respectively, are investigated. It is proved that the dimension of ϕ_n^i (i=1,2) is equal to (p^n+(-1)^i )/2, all weight multiplicities of these representations are equal to 1, their restrictions to the group G_k naturally embedded into G_n are completely reducible with irreducible constituents ϕ_k^1 and ϕ_k^2, and their restrictions to Sp_2n(p) can be obtained as the result of the reduction modulo p of certain complex irreducible representations of the group Sp_2n(p).

Journal of Group Theory, 2000

For k = 1, 2, ..., n−1 let V k = V (λ k ) be the Weyl module for the special orthogonal group G = SO(2n + 1, F) with respect to the k-th fundamental dominant weight λ k of the root system of type Bn and put Vn = V (2λn). It is well known that all of these modules are irreducible when char(F) = 2 while when char(F) = 2 they admit many proper submodules. In this paper, assuming that char(F) = 2, we prove that V k admits a chain of

Journal of Algebraic Combinatorics, 2013

Given a non-singular quadratic form q on V := V (2n + 1, F), let ∆ be the building of type Bn formed by the subspaces of V totally singular for q and, for 1 ≤ k ≤ n, let ∆ k be the k-grassmannian of ∆. Let ε k be the embedding of ∆ k in PG( k V ) mapping every point v1, v2, ..., v k of ∆ k to the point v1 ∧v2 ∧...∧v k of PG( k V ). It is known that if char(F) = 2 then dim(ε k ) = 2n+1 k . In this paper we give a new very easy proof of this fact. We also prove that if char(F) = 2 then dim(ε k ) = 2n+1 k − 2n+1 k−2 . As a consequence, when 1 < k < n and char(F) = 2 the embedding ε k is not universal. Finally, we prove that if F is a perfect field of characteristic p > 2 or a number field, n > k and k = 2 or 3, then ε k is universal.

European Journal of Combinatorics, 2009

Journal of Algebra, 2006

Let G be a finite group and χ be an irreducible character. We say that a subgroup H is a χ -subgroup if the restriction χ H of χ to H has at least one linear constituent of multiplicity 1. Not every pair (G, χ ) has a χ -subgroup, but χ -subgroups can be found in many cases. The existence of such subgroups is of interest for several reasons, one being that knowledge of a χ -subgroup enables us to give a simple construction of a matrix representation of G affording χ . In this paper we show that, when G = Sp(4, q) where q is a power of an odd prime p and H is a Sylow p-subgroup of G, then H is a χ -subgroup for every irreducible character χ (with one exception). We also find a p-subgroup which is a χ -subgroup for the exceptional character.

Simon Stevin, 2006

European Journal of Combinatorics, 2007

Exploiting the interplay between hyperbolic and isotropic geometry, we prove that the grassmannian of totally isotropic k-spaces of the polar space associated to the symplectic group Sp 2n (F) has generating rank 2n k − 2n k−2 when Char(F) = 2.

We prove that the grassmannian of lines of the polar space associated to Sp2n(F) has generating rank 2n2 n 1 when Char(F) 6= 2. MSC 2000: 51E24 (primary); 51A50, 51A45 (secondary)

Journal of Pure and Applied Algebra, 2013

Let W be a vector space over an algebraically closed field k. Let H be a quasisimple group of Lie type of characteristic p = char(k) acting irreducibly on W . Suppose that G is a classical group with natural module W . Suppose also that G is a classical group with natural module, chosen minimally with respect to containing the image of H under the associated representation. We consider the question of when H can act irreducibly on a G constituent of W ⊗e and study its relationship to the maximal subgroup problem for finite classical groups.

2021

The irreducible representations φn and φ 2 n of the symplectic group Gn = Sp2n(P ) over an algebraically closed field P of characteristic p &gt; 2 with highest weights ωn−1 + p−3 2 ωn and p−1 2 ωn, respectively, are investigated. It is proved that the dimension of φn (i = 1, 2) is equal to (p+(−1))/2, all weight multiplicities of these representations are equal to 1, their restrictions to the group Gk naturally embedded into Gn are completely reducible with irreducible constituents φk and φ 2 k, and their restrictions to Sp2n(p) can be obtained as the result of the reduction modulo p of certain complex irreducible representations of the group Sp2n(p). These results allow us to obtain the exact list of rational irreducible representations of simple algebraic groups over fields of positive characteristics all whose weight subspaces have dimension 1. This generalizes a result of Seitz. In this paper we consider the irreducible representations of the symplectic group Sp2n(P ) over an al...

Journal of Combinatorial Theory, Series A, 2020

In this paper we compute the dimension of the Grassmann embeddings of the polar Grassmannians associated to a possibly degenerate Hermitian, alternating or quadratic form with possibly nonmaximal Witt index. Moreover, in the characteristic 2 case, when the form is quadratic and non-degenerate with bilinearization of minimal Witt index, we define a generalization of the so-called Weyl embedding (see [4]) and prove that the Grassmann embedding is a quotient of this generalized 'Weyl-like' embedding. We also estimate the dimension of the latter.

2005

In this note we use Bott-Borel-Weil theory to compute cohomology of interesting vector bundles on sequences of Grassmanians.