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Fabio Scarabotti

Publications and source records attributed to Fabio Scarabotti.

12 recordsLinked to original sources
arXiv2025

Representation theory of the group of automorphisms of a finite rooted tree

We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions, a natural generalization of set compositions but with new features and more complexity. These combinatorial structures lead to a family of permutation representations which have the same parametrization of the irreducible representations. Our trees are not necessarily spherically homogeneous and our approach is coordinate free.

math.RT↗
arXiv2024

Homogeneous spaces of semidirect products and finite Gelfand pairs

Let $K\leq H$ be two finite groups and let $C\leq A$ be two finite abelian groups, with $H$ acting on $A$ as a group of isomorphisms admitting $C$ as a $K$-invariant subgroup. We study the homogeneous space $X\coloneqq\left(H\ltimes A\right)/\left(K\ltimes C\right)$ and determine the decomposition of the permutation representation of $H\ltimes A$ acting on $X$. We then characterize when this is multiplicity-free, that is, when $\left(H\ltimes A,K\ltimes C\right)$ is a Gelfand pair. If this is the case, we explicitly calculate the corresponding spherical functions. From our general construction and related analysis, we recover Dunkl's results on the $q$-analog of the nonbinary Johnson scheme.

math.RT↗
arXiv2018

Harmonic analysis and spherical functions for multiplicity-free induced representations of finite groups

In this work, we study multiplicity-free induced representations of finite groups. We analyze in great detail the structure of the Hecke algebra corresponding to the commutant of an induced representation and then specialize to the multiplicity-free case. We then develop a suitable theory of spherical functions that, in the case of induction of the trivial representation of the subgroup, reduces to the classical theory of spherical functions for finite Gelfand pairs. \par We also examine in detail the case when we induce from a normal subgroup, showing that the corresponding harmonic analysis can be reduced to that on a suitable Abelian group. \par The second part of the work is devoted to a comprehensive study of two examples constructed by means of the general linear group GL$(2,\mathbb{F}_q)$, where $\mathbb{F}_q$ is the finite field on $q$ elements. In the first example we induce an indecomposable character of the Cartan subgroup. In the second example we induce to GL$(2,\mathbb{F}_{\!q^2})$ a cuspidal representation of GL$(2,\mathbb{F}_q)$.

math.RT↗
arXiv2014

Mackey's theory of $τ$-conjugate representations for finite groups. APPENDIX: On Some Gelfand Pairs and Commutative Association Schemes

The aim of the present paper is to expose two contributions of Mackey, together with a more recent result of Kawanaka and Matsuyama, generalized by Bump and Ginzburg, on the representation theory of a finite group equipped with an involutory anti-automorphism (e.g. the anti-automorphism $g\mapsto g^{-1}$). Mackey's first contribution is a detailed version of the so-called Gelfand criterion for weakly symmetric Gelfand pairs. Mackey's second contribution is a characterization of simply reducible groups (a notion introduced by Wigner). The other result is a twisted version of the Frobenius-Schur theorem, where "twisted" refers to the above-mentioned involutory anti-automorphism. APPENDIX: We consider a special condition related to Gelfand pairs. Namely, we call a finite group $G$ and its automorphism $σ$ satisfy Condition ($\bigstar$) if the following condition is satisfied: if for $x,y\in G$, $x\cdot x^{-σ}$ and $y\cdot y^{-σ}$ are conjugate in $G$, then they are conjugate in $K=C_G(σ)$. We study the meanings of this condition, as well as showing many examples of $G$ and $σ$ which do (or do not) satisfy Condition ($\bigstar$).

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arXiv2014

Mackey's criterion for subgroup restriction of Kronecker products and harmonic analysis on Clifford groups

We present a criterion for multiplicity-freeness of the decomposition of the restriction Res$^G_H(ρ_1 \otimes ρ_2)$ of the Kronecker product of two generic irreducible representations $ρ_1, ρ_2$ of a finite group $G$ with respect to a subgroup $H \leq G$. This constitutes a generalization of a well known criterion due to Mackey (which corresponds to the case $H = G$). The corresponding harmonic analysis is illustated by detailed computations on the Clifford groups $G={\mathbb{CL}}(n)$, together with the subgroups $H={\mathbb{CL}}(n-1)$, for $n \geq 1$, which lead to an explicit decomposition of the restriction of Kronecker products.

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arXiv2013

Induced representations and harmonic analysis on finite groups

Given a finite group $G$ and a subgroup $K$, we study the commutant of $\text{Ind}_K^Gθ$, where $θ$ is an irreducible $K$-representation. After a careful analysis of Frobenius reciprocity, we are able to introduce an orthogonal basis in such commutant and an associated Fourier transform. Then we translate our results in the corresponding Hecke algebra, an isomorphic algebra in the group algebra of $G$. Again a complete Fourier analysis is developed and, as particular cases, we obtain some results of Curtis and Fossum on the irreducible characters of Hecke algebras. Finally, we develop a theory of Gelfand-Tsetlin bases for Hecke algebras.

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arXiv2011

A factorization method for q-Racah polynomials

We develop a factorization method for q-Racah polynomials. It is inspired to the approach to q-Hahn polynomials based on the q-Johnson scheme but we do not use association scheme theory nor Gel'fand pairs, but only manipolation of q-difference operators.

math.CA↗
arXiv2009

The tree method for multidimensional q-Hahn and q-Racah polynomials

We develop a tree method for multidimensional q-Hahn polynomials. We define them as eigenfunctions of a multidimensional q-difference operator and we use the factorization of this operator as a key tool. Then we define multidimensional q-Racah polynomials as the connection coefficients between different bases of q-Hahn polynomials. We show that our multidimensional q-Racah polynomials may be expressed as product of ordinary one-dimensional q-Racah polynomial by means of a suitable sequence of transplantations of edges of the trees.

math.CA↗
arXiv2008

Multidimensional Hahn polynomials, intertwining functions on the symmetric group and Clebsch-Gordan coefficients

We generalize a construction of Dunkl, obtaining a wide class intertwining functions on the symmetric group and a related family of multidimensional Hahn polynomials. Following a suggestion of Vilenkin and Klymik, we develop a tree-method approach for those intertwining functions. We also give a group theoretic proof of the relation between Hahn polynomials and Clebesh-Gordan coefficients, given analytically by Koornwinder and by Nikiforov, Smorodinskiĭ and Suslov. Such relation is also extended to the multidimensional case.

math.CA↗
arXiv2007

Harmonic analysis of finite lamplighter random walks

Recently, several papers have been devoted to the analysis of lamplighter random walks, in particular when the underlying graph is the infinite path $\mathbb{Z}$. In the present paper, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the $C_2$-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph. In the case the graph has a transitive isometry group $G$, we also describe the spectral analysis in terms of the representation theory of the wreath product $C_2\wr G$. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples were already studied by Haggstrom and Jonasson by probabilistic methods.

math.PR↗
arXiv2007

Harmonic analysis on a finite homogeneous space

In this paper, we study harmonic analysis on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity. After a careful look at Frobenius reciprocity and transitivity of induction, and the introduction of three types of spherical functions, we develop a theory of Gelfand Tsetlin bases for permutation representations. Then we study several concrete examples on the symmetric groups, generalizing the Gelfand pair of the Johnson scheme; we also consider statistical and probabilistic applications. After that, we consider the composition of two permutation representations, giving a non commutative generalization of the Gelfand pair associated to the ultrametric space; actually, we study the more general notion of crested product. Finally, we consider the exponentiation action, generalizing the decomposition of the Gelfand pair of the Hamming scheme; actually, we study a more general construction that we call wreath product of permutation representations, suggested by the study of finite lamplighter random walks. We give several examples of concrete decompositions of permutation representations and several explicit 'rules' of decomposition.

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