arXiv subjects

Mihalis Maliakas

Publications and source records attributed to Mihalis Maliakas.

13 recordsLinked to original sources

arXiv2026

Tableaux and orbit harmonics quotients for finite transformation monoids

We extend Grood's tableau construction of irreducible representations of the rook monoid and Steinberg's analogous result for the full transformation monoid. Our approach is characteristic-free and applies to any submonoid $\mathcal{M}(n)$ of the partial transformation monoid on an $n$-element set that contains the symmetric group. To achieve this, we introduce and study a functor from the category of rational representations of the monoid of $n \times n$ matrices to the category of finite dimensional representations of $\mathcal{M}(n)$. We establish two branching rules. Our main results describe graded module structures of orbit harmonics quotients for the rook, partial transformation, and full transformation monoids. This yields analogs of the Cauchy decomposition for polynomial rings in $n\times n$ variables.

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arXiv2026

On the free LAnKe on $3n-2$ generators: a theorem of Friedmann, Hanlon, Stanley and Wachs

A LAnKe (also known as a Filippov algebra or a Lie algebra of the $n$-th kind) is a vector space equipped with a skew-symmetric $n$-linear form that satisfies the generalized Jacobi identity. Friedmann, Hanlon, Stanley and Wachs have shown that the symmetric group acts on the multilinear part of the free LAnKe on $2n-1$ generators as an irreducible representation. They announced that the multilinear component on $3n-2$ generators decomposes as a direct sum of two irreducible symmetric group representations and a proof was given recently in a subsequent paper by Friedmann, Hanlon and Wachs. In the present paper we provide a proof of the later statement. The two proofs are substantially different.

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arXiv2025

Total trades, intersection matrices and Specht modules

Trades are important objects in combinatorial design theory that may be realized as certain elements of kernels of inclusion matrices. Total trades were introduced recently by Ghorbani, Kamali and Khosravshahi, who showed that over a field of characteristic zero the vector space of trades decomposes into a direct sum of spaces of total trades. In this paper, we show that the vector space spanned by the permutations of a total trade is an irreducible representation of the symmetric group. As a corollary, the previous decomposition theorem is recovered. Also, a basis is obtained for the module of total trades in the spirit of Specht polynomials. More generally, in the second part of the paper we consider intersection matrices and determine the irreducible decompositions of their images. This generalizes previously known results concerning ranks of special cases.

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arXiv2025

On the action of the symmetric group on the free LAnKe: a question of Friedmann, Hanlon, Stanley and Wachs

A LAnKe (also known as a Lie algebra of the $n$th kind, or a Filippov algebra) is a vector space equipped with a skew-symmetric $n$-linear form that satisfies the generalized Jacobi identity. The symmetric group $\mathfrak{S}_m$ acts on the multilinear part of the free LAnKe on $m=(n-1)k+1$ generators, where $k$ is the number of brackets, by permutation of the generators. The corresponding representation was studied by Friedmann, Hanlon, Stanley and Wachs, who asked whether for $n \ge k$, its irreducible decomposition contains no summand whose Young diagram has at most $k-1$ columns. The answer is affirmative if $k \le 3$. In this paper, we show that the answer is affirmative for all $k$. A proof has been given recently by Friedmann, Hanlon and Wachs. The two proofs are completely different.

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arXiv2024

Presentations of Schur and Specht modules in characteristic zero

New presentations of Specht modules of symmetric groups over fields of characteristic zero have been obtained by Brauner, Friedmann, Hanlon, Stanley and Wachs. These involve generators that are column tabloids and relations that are Garnir relations with maximal number of exchanges between consecutive columns or symmetrization of Garnir relations with minimal number of exchanges between consecutive columns. In this paper, we examine Garnir relations and their symmetrization with any number of exchanges. In both cases, we provide sufficient arithmetic conditions so that the corresponding quotient is a Specht module. In particular, in the first case this yields new presentations of Specht modules if the parts of the conjugate partition that correspond to maximal number of exchanges greater than 1 are distinct. These results generalize the presentations mentioned above and offer an answer to a question of Friedmann, Hanlon and Wachs. Our approach is via representations of the general linear group.

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arXiv2024

On stability and nonvanishing of homomorphism spaces between Weyl modules

Consider the general linear group $G=GL_{n}(K)$ defined over an infinite field $K$ of positive characteristic $p$. We denote by $Δ(λ)$ the Weyl module of $G$ which corresponds to a partition $λ$. Let $λ, μ$ be partitions of $r$ and let $γ$ be partition with all parts divisible by $p$. In the first main result of this paper, we find sufficient conditions on $λ, μ$ and $γ$ so that $Hom_G(Δ(λ),Δ(μ))$ $ \simeq$ $ Hom_G(Δ(λ+γ),Δ(μ+γ))$, thus providing an answer to a question of D. Hemmer. As corollaries we obtain stability and periodicity results for homomorphism spaces. In the second main result we find related sufficient conditions on $λ, μ$ and $p$ so that $Hom_G(Δ(λ),Δ(μ))$ is nonzero. An explicit map is provided that corresponds to the sum of all semistandard tableaux of shape $μ$ and weight $λ$.

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arXiv2024

Relating homomorphism spaces between Specht modules of different degrees

Let $K$ be an infinite field of characteristic $p>0$ and let $λ, μ$ be partitions of $n$, where $λ=(λ_1,...,λ_n)$ and $μ=(μ_1,..,μ_n)$. By $S^λ$ we denote the Specht module corresponding to $λ$ for the group algebra $K\mathfrak{S}_n$ of the symmetric group $\mathfrak{S}_n$. D. Hemmer has raised the question of relating the homomorphism spaces $\Hom_{\mathfrak{S}_n}(S^μ, S^λ)$ and $\Hom_{\mathfrak{S}_{n'}}(S^{μ^+}, S^{λ^+})$, where $n'=n+kp^d$, $λ^+ =λ+(kp^{d})$, $μ^+=μ+(kp^{d})$, and $d, k$ are positive integers. We show that these are isomorphic if $p$ is odd, $p^d >\min\{λ_2, μ_1-λ_1\}$ and $μ_2 \le λ_1$.

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arXiv2024

A periodicity theorem for extensions of Weyl modules

In this paper we study periodicity phenomena for modular extensions between Weyl modules and between Weyl and simple modules of the general linear group that are associated to adding a power of the characteristic to the first parts of the involved partitions.

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arXiv2021

On extensions of hook Weyl modules

We determine the integral extension groups $Ext^1(Δ(h),Δ(h(k)))$ and $Ext^k(Δ(h),Δ(h(k)))$, where $Δ(h),Δ(h(k))$ are the Weyl modules of the general linear group $GL_n$ corresponding to the hook partitions $h=(a,1^b)$, $h(k)=(a+k,1^{b-k})$.

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arXiv2021

On homomorphisms into Weyl modules corresponding to partitions with two parts

Let $K$ be an infinite field of characteristic $p>0$ and let $λ, μ$ be partitions, where $μ$ has two parts. We find sufficient arithmetic conditions on $p, λ, μ$ for the existence of a nonzero homomorphism $Δ(λ) \to Δ(μ)$ of Weyl modules for the general linear group $GL_n(K)$. Also for each $p$ we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.

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arXiv2021

On homomorphisms involving a hook Weyl module

Let $k$ be an infinite field of positive characteristic. We determine all homomorphisms between Weyl modules for $GLn(k)$, where one of the partitions is a hook. As a consequence we obtain a nonvanishing result concerning homomorphisms between Weyl modules for algebraic groups of type B, C and D when one of the partitions is a hook.

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arXiv2008

On Weyl resolutions associated to Frobenius twists

We construct Weyl resolutions associated to certain Frobenius twists of divided powers. Using these and other related complexes we obtain the Weyl filtration dimension of the Schur algebras S(2,r), a result due to A. Parker.

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arXiv2006

On the Divided Power Algebra and the Symplectic Group in Chracteristic 2

Let V be an even dimensional vector space over a field K of characteristic 2 equipped with a non-degenerate alternating bilinear form f. The divided power algebra DV is considered as a complex with differential defined from f. We examine the cohomology modules as representations of the corresponding symplectic group.

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