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Anthony Giaquinto

Publications and source records attributed to Anthony Giaquinto.

At least 19 recordsLinked to original sources
arXiv2026

Fence Posets, Good Gradings and Frobenius Maximal Parabolics

Let $\mathfrak L$ be a Frobenius maximal parabolic subalgebra of $\mathfrak{sl}_n$. For any $F\in\mathfrak L^*$ for which the Kirillov form $B_F(x,y)=F([x,y])$ is non-degenerate, let $\widehat F$ denote the associated principal element. We prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak L$ form a unimodal sequence symmetric about $\frac12$. We also prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak{gl}_n$ form a unimodal sequence symmetric about $0$. The proof relates the ranked meander associated to $\mathfrak L$ to the order ideals of a related fence poset through Panyushev reduction. The known unimodality of the rank polynomial of the fence poset implies that of the meander, which in turn determines a good grading of $\mathfrak{gl}_n$ in the sense of Elashvili and Kac. We prove that this grading coincides with that induced by the principal element and that the pyramid associated to this grading may be filled in such a way that its good element $e$ lies in $\mathfrak L$. The two unimodality results then follow from the injectivity properties of $\operatorname{ad}_e$ coming from the good grading and the duality induced by the bilinear form $B_F$.

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arXiv2026

The Center of the Temperley-Lieb Algebra

We compute the dimension of the center of the Temperley--Lieb algebra $\operatorname{TL_n}(\delta)$ over a field of characteristic zero for every nonzero value of the parameter $\delta$. The proof uses the cellular filtration by cup number, together with known facts about the representation theory of the Temperley--Lieb algebra, especially the structure of its standard modules and their radicals. Dilation and compression maps compare the induced graded pieces of the center at levels $n$ and $n-2$, giving an upper bound of one for each such piece. A deformation argument gives the matching lower bound, and hence $ \dim Z(\operatorname{TL}_n(\delta))=1+\Bigl\lfloor \frac{n}{2}\Bigr\rfloor$. We also prove that every central element is fixed by the canonical anti-automorphism and by the natural diagram-reflection automorphism. Finally, we give a congruence criterion for the trivial-radical case and record a Gram-matrix computation for leading terms.

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arXiv2026

Lowering operators, orthogonal decomposition of tensor space, and quantized Schur--Weyl duality

For $q$ generic, Jimbo showed that $q$-tensor space $V_q^{\otimes r}$ (where $V_q$ is the $n$-dimensional vector representation) satisfies Schur--Weyl duality with respect to the commuting actions of the quantized enveloping algebra $\mathbf{U}_q(\mathfrak{gl}_n)$ and the Iwahori--Hecke algebra $\mathbf{H}_q(\mathfrak{S}_r)$, with the latter action derived from the $R$-matrix. In the limit as $q \to 1$, one recovers classical Schur--Weyl duality. Using a recursive construction of certain linear combinations $Ψ_j$ of Coxeter monomials in the negative part of $\mathbf{U}_q(\mathfrak{gl}_n)$, we give a combinatorial realization of the corresponding isotypic semisimple decomposition of $V_q^{\otimes r}$, indexed by paths in the Bratteli diagram. This extends earlier work (Journal of Algebra 2024) of the first two authors for the case $n =2$. Our construction works over any field containing a non-zero element $q$ which is not a root of unity. The element $Ψ_j$ depends on a weight $λ$ and is the ``evaluation at $λ$'' of a certain $q$-lowering operator $\overlineΨ_j$ satisfying a similar recursion, up to renormalization. This simplifies the construction of lowering operators. Both $Ψ_j$ and $\overlineΨ_j$ are independent of a choice of root vectors. On the other hand, the $Ψ_j$ can be applied to construct root vectors (independent of the braid group action) as explicit linear combinations of Coxeter monomials.

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arXiv2023

An orthogonal realization of representations of the Temperley-Lieb algebra

Under a suitable hypothesis, we construct a full set of pairwise orthogonal maximal vectors in $V^{\otimes n}$, where $V=V(1)$ is the simple module of highest weight $1$ for the quantized enveloping algebra $\mathbf{U}(\mathfrak{sl}_2)$. We give a number of applications, one of which is an orthogonal basis of the simple modules for the Temperley-Lieb algebra $\text{TL}_n$. We relate this new orthogonal basis to the standard cellular basis.

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arXiv2022

The partial Temperley-Lieb algebra and its representations

We give a combinatorial description of a new diagram algebra, the partial Temperley--Lieb algebra, arising as the generic centralizer algebra $\mathrm{End}_{\mathbf{U}_q(\mathfrak{gl}_2)}(V^{\otimes k})$, where $V = V(0) \oplus V(1)$ is the direct sum of the trivial and natural module for the quantized enveloping algebra $\mathbf{U}_q(\mathfrak{gl}_2)$. It is a proper subalgebra of the Motzkin algebra (the $\mathbf{U}_q(\mathfrak{sl}_2)$-centralizer) of Benkart and Halverson. We prove a version of Schur--Weyl duality for the new algebras, and describe their generic representation theory.

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arXiv2022

Schur--Weyl duality for twin groups

The twin group $TW_n$ on $n$ strands is the group generated by $t_1, \dots, t_{n-1}$ with defining relations $t_i^2=1$, $t_it_j = t_jt_i$ if $|i-j|>1$. We find a new instance of semisimple Schur--Weyl duality for tensor powers of a natural $n$-dimensional reflection representation of $TW_n$, depending on a parameter $q$. At $q=1$ the representation coincides with the natural permutation representation of the symmetric group, so the new Schur--Weyl duality may be regarded as a $q$-analogue of the one motivating the definition of the partition algebra.

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arXiv2021

Schur-Weyl duality for tensor powers of the Burau representation

Artin's braid group $B_n$ is generated by $σ_1, \dots, σ_{n-1}$ subject to the relations \[ σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1}, \quad σ_iσ_j = σ_j σ_i \text{ if } |i-j|>1. \] For complex parameters $q_1,q_2$ such that $q_1q_2 \ne 0$, the group $B_n$ acts on the vector space $\mathbf{E} = \sum_i \mathbb{C} \mathbf{e}_i$ with basis $\mathbf{e}_1, \dots, \mathbf{e}_n$ by \begin{gather*} σ_i \cdot \mathbf{e}_i = (q_1+q_2)\mathbf{e}_i + q_1\mathbf{e}_{i+1}, \quad σ_i \cdot \mathbf{e}_{i+1} = -q_2\mathbf{e}_i, \\ σ_i \cdot \mathbf{e}_j = q_1 \mathbf{e}_j \text{ if } j \ne i,i+1. \end{gather*} This representation is (a slight generalization of) the Burau representation. If $q = -q_2/q_1$ is not a root of unity, we show that the algebra of all endomorphisms of $\mathbf{E}^{\otimes r}$ commuting with the $B_n$-action is generated by the place-permutation action of the symmetric group $S_r$ and the operator $p_1$, given by \[ p_1(\mathbf{e}_{j_1} \otimes \mathbf{e}_{j_2} \otimes \cdots \otimes \mathbf{e}_{j_r}) = q^{j_1-1} \, \sum_{i=1}^n \mathbf{e}_i \otimes \mathbf{e}_{j_2} \otimes \cdots \otimes \mathbf{e}_{j_r} . \] Equivalently, as a $(\mathbb{C} B_n, \mathcal{P}'_r([n]_q))$-bimodule, $\mathbf{E}^{\otimes r}$ satisfies Schur--Weyl duality, where $\mathcal{P}'_r([n]_q)$ is a certain subalgebra of the partition algebra $\mathcal{P}_r([n]_q)$ on $2r$ nodes with parameter $[n]_q = 1+q+\cdots + q^{n-1}$, isomorphic to the semigroup algebra of the "rook monoid" studied by W. D. Munn, L. Solomon, and others.

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arXiv2019

Peter-Weyl bases, preferred deformations, and Schur-Weyl duality

We discuss the deformed function algebra of a simply connected reductive Lie group G over the complex numbers using a basis consisting of matrix elements of finite dimensional representations. This leads to a preferred deformation, meaning one where the structure constants of comultiplication are unchanged. The structure constants of multiplication are controlled by quantum 3j symbols. We then discuss connections earlier work on preferred deformations that involved Schur-Weyl duality.

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arXiv2014

Cellular bases of generalized q-Schur algebras

We show that cellular bases of generalized $q$-Schur algebras can be constructed by gluing arbitrary bases of the cell modules and their dual basis (with respect to the anti-involution giving the cell structure) along defining idempotents. For the rational form, over the field $\mathbb{Q}(v)$ of rational functions in an indeterminate $v$, our proof of this fact is self-contained and independent of the theory of quantum groups. In the general case, over a commutative ring $\Bbbk$ regarded as a $\mathbb{Z}[v,v^{-1}]$-algebra via specialization $v \mapsto q$ for some chosen invertible $q \in \Bbbk$, our argument depends on the existence of the canonical basis.

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arXiv2012

Deformations associated to rigid algebras

The deformations of an infinite dimensional algebra may be controlled not just by its own cohomology but by that of an associated diagram of algebras, since an infinite dimensional algebra may be absolutely rigid in the classical deformation theory for single algebras while depending essentially on some parameters. Two examples studied here, the function field of a sphere with four marked points and the first Weyl algebra, show, however, that the existence of these parameters may be made evident by the cohomology of a diagram (presheaf) of algebras constructed from the original. The Cohomology Comparison Theorem asserts, on the other hand, that the cohomology and deformation theory of a diagram of algebras is always the same as that of a single, but generally rather large, algebra constructed from the diagram.

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arXiv2012

On the cohomology of the Weyl algebra, the quantum plane, and the q-Weyl algebra

Deformation theory can be used to compute the cohomology of a deformed algebra with coefficients in itself from that of the original. Using the invariance of the Euler-Poincare characteristic under deformation, it is applied here to compute the cohomology of the Weyl algebra, the algebra of the quantum plane, and the q-Weyl algebra. The behavior of the cohomology when q is a root of unity may encode some number theoretic information.

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arXiv2011

Meander graphs and Frobenius Seaweed Lie algebras

The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine this graph in several ways with a goal of determining families of Frobenius (index zero) seaweed algebras. Our analysis gives two new families of Frobenius seaweed algebras as well as elementary proofs of known families of such Lie algebras.

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arXiv2010

Topics in Algebraic Deformation Theory

We give a selective survey of topics in algebraic deformation theory ranging from its inception to current times. Throughout, the numerous contributions of Murray Gerstenhaber are emphasized, especially the common themes of cohomology, infinitesimal methods, and explicit global deformation formulas.

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arXiv2009

The Principal Element of a Frobenius Lie Algebra

We introduce the notion of the \textit{principal element} of a Frobenius Lie algebra $\f$. The principal element corresponds to a choice of $F\in \f^*$ such that $F[-,-]$ non-degenerate. In many natural instances, the principal element is shown to be semisimple, and when associated to $\sl_n$, its eigenvalues are integers and are independent of $F$. For certain ``small'' functionals $F$, a simple construction is given which readily yields the principal element. When applied to the first maximal parabolic subalgebra of $\sl_n$, the principal element coincides with semisimple element of the principal three-dimensional subalgebra. We also show that Frobenius algebras are stable under deformation.

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arXiv2008

Graphs, Frobenius functionals, and the classical Yang-Baxter equation

A Lie algebra is Frobenius if it admits a linear functional F such that the Kirillov form F([x,y]) is non-degenerate. If g is the m-th maximal parabolic subalgebra P(n,m) of sl(n) this occurs precisely when (n,m) = 1. We define a "cyclic" functional F on P(n,m) and prove it is non-degenerate using properties of certain graphs associated to F. These graphs also provide in some cases readily computable associated solutions of the classical Yang-Baxter equation. We also define a local ring associated to each connected loopless graph from which we show that the graph can be reconstructed. Finally, we examine the seaweed Lie algebras of Dergachev and Kirillov from our perspective.

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arXiv2005

Presenting generalized Schur algebras in types B,C,D

We give explicit presentations by generators and relations of certain generalized Schur algebras (associated with tensor powers of the natural representation) in types B, C, D. This extends previous results in type A obtained by two of the authors. The presentation is compatible with the Serre presentation of the corresponding universal enveloping algebra. In types C, D this gives a presentation of the corresponding classical Schur algebra (the image of the representation on a tensor power) since the classical Schur algebra coincides with the generalized Schur algebra in those types. This coincidence between the generalized and classical Schur algebra fails in type B, in general.

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arXiv2003

Algebraic deformations arising from orbifolds with discrete torsion

We develop methods for computing Hochschild cohomology groups and deformations of crossed product rings. We use these methods to find deformations of a ring associated to a particular orbifold with discrete torsion, and give a presentation of the center of the resulting deformed ring. This connects with earlier calculations by Vafa and Witten of chiral numbers and deformations of a similar orbifold.

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