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Matan Seidel

Publications and source records attributed to Matan Seidel.

6 recordsLinked to original sources
arXiv2025

Exposure Orders in Free Group Algebras: Minimal Schreier Transversals, Free Bases, and Gr\"obner Bases

Consider the free group algebra $K\left[F\right]$, where $F$ is a free group and $K$ a field. A well-order $\prec$ on $F$ is called an exposure order if words are greater than their proper prefixes. We show that every one-sided ideal $I$ in $K\left[F\right]$ admits a Schreier transversal, a basis, and a Gr\"obner basis -- each minimal in a natural sense with respect to $\prec$. When $I$ is finitely generated and $\prec$ is computable, we provide an algorithm for computing these minimal structures from a finite generating set. This extends the foundational works of Lewin and Rosenmann, which relied on the shortlex order, to both a broader class of orders on $F$ and to infinitely generated ideals, while retaining algorithmic capabilities for such orders in case $I$ is finitely generated. General exposure orders lack a form of compatibility with products which we call suffix-invariance, that shortlex enjoys, and which prior Gr\"obner basis constructions in $K\left[F\right]$ relied on. In its absence, reductions may strictly increase the support of elements, requiring nontrivial conceptual adaptations to definitions and algorithms. These adaptations clarify the notion of minimality underlying prior constructions and demonstrate that algorithmic Gr\"obner theory in $K\left[F\right]$ does not fundamentally require suffix-invariance, although its presence -- as in shortlex -- results in a simpler theory. Our framework further illuminates the flexibility of exposure orders: with a suitable choice of $\prec$, any Schreier transversal for $I$ can be realized as minimal, and any basis for $I$ arising from the constructions of Lewin or Rosenmann can likewise be realized as its minimal basis, unifying both approaches under a single framework.

math.GR↗
arXiv2025

Primitivity Testing in Free Group Algebras via Duality

Let $K$ be a field and $F$ a free group. By a classical result of Cohn and Lewin, the free group algebra $K\left[F\right]$ is a free ideal ring (FIR): a ring over which the submodules of free modules are themselves free, and of a well-defined rank. Given a finitely generated right ideal $I\leq K\left[F\right]$ and an element $f\in I$, we give an explicit algorithm determining whether $f$ is part of some basis of $I$. More generally, given free $K[F]$-modules $M\le N$, we provide algorithms determining whether $M$ is a free summand of $N$, and whether $N$ admits a free splitting relative to $M$. These can also be used to obtain analogous algorithms for free groups $H\le J$. As an aside, we also provide an algorithm to compute the intersection of two given submodules of a free $K\left[F\right]$-module. A key feature of this work is the introduction of a duality, induced by a matrix with entries in a free ideal ring, between the respective algebraic extensions of its column and row spaces.

math.GR↗
arXiv2023

Stable Invariants of Words from Random Matrices

Let $w$ be a word in a free group. A few years ago, Magee and the first named author discovered that the stable commutator length (scl) of $w$, a well-known topological invariant, can also be defined in terms of certain Fourier coefficients of $w$-random unitary matrices [arXiv:1802.04862]. But the random-matrix side of this equality can be naturally tweaked by considering $w$-random permutations, $w$-random orthogonal matrices and so on, to produce new invariants for any given word. Are these invariants new? interesting? Do they admit an intrinsic topological description as in the case of $w$-random unitaries and scl? The current paper formalizes the definition of these invariants coming from $w$-random matrices, answers the above questions in certain cases involving generalized symmetric groups, and poses detailed conjectures in many others. In particular, we present a plethora of topological, combinatorial and algebraic invariants of words which play, or are at least conjectured to play, a similar role to the one played by scl in the above-mentioned result. Among others, these invariants include two invariants recently defined by Wilton [arXiv:2210.09853]: the stable primitivity rank and a non-oriented analog of scl.

math.GR↗
arXiv2021

Word Measures on $GL_N(q)$ and Free Group Algebras

Fix a finite field $K$ of order $q$ and a word $w$ in a free group $F$ on $r$ generators. A $w$-random element in $GL_N(K)$ is obtained by sampling $r$ independent uniformly random elements $g_1,\ldots,g_r\in GL_N(K)$ and evaluating $w\left(g_1,\ldots,g_r\right)$. Consider $\mathbb{E}_w\left[\mathrm{fix}\right]$, the average number of vectors in $K^{N}$ fixed by a $w$-random element. We show that $\mathbb{E}_{w}\left[\mathrm{fix}\right]$ is a rational function in $q^{N}$. Moreover, if $w=u^{d}$ with $u$ a non-power, then the limit $\lim_{N\to\infty}\mathbb{E}_{w}\left[\mathrm{fix}\right]$ depends only on $d$ and not on $u$. These two phenomena generalize to all stable characters of the groups $\left\{ GL_N(K)\right\}_{N}$. A main feature of this work is the connection we establish between word measures on $GL_N(K)$ and the free group algebra $K\left[F\right]$. A classical result of Cohn [1964] and Lewin [1969] is that every one-sided ideal of $K\left[F\right]$ is a free $K\left[F\right]$-module with a well-defined rank. We show that for $w$ a non-power, $\mathbb{E}_{w}\left[\mathrm{fix}\right]=2+\frac{C}{q^{N}}+O\left(\frac{1}{q^{2N}}\right)$, where $C$ is the number of rank-2 right ideals $I\le K\left[F\right]$ which contain $w-1$ but not as a basis element. We describe a full conjectural picture generalizing this result, featuring a new invariant we call the $q$-primitivity rank of $w$. In the process, we prove several new results about free group algebras. For example, we show that if $T$ is any finite subtree of the Cayley graph of $F$, and $I\le K\left[F\right]$ is a right ideal with a generating set supported on $T$, then $I$ admits a basis supported on $T$. We also prove an analogue of Kaplansky's unit conjecture for certain $K\left[F\right]$-modules.

math.GR↗
arXiv2020

Topological obstructions to quantum computation with unitary oracles

Algorithms with unitary oracles can be nested, which makes them extremely versatile. An example is the phase estimation algorithm used in many candidate algorithms for quantum speed-up. The search for new quantum algorithms benefits from understanding their limitations: Some tasks are impossible in quantum circuits, although their classical versions are easy, for example, cloning. An example with a unitary oracle $U$ is the if clause, the task to implement controlled $U$ (up to the phase on $U$). In classical computation the conditional statement is easy and essential. In quantum circuits the if clause was shown impossible from one query to $U$. Is it possible from polynomially many queries? Here we unify algorithms with a unitary oracle and develop a topological method to prove their limitations: No number of queries to $U$ and $U^\dagger$ lets quantum circuits implement the if clause, even if admitting approximations, postselection and relaxed causality. We also show limitations of process tomography, oracle neutralization, and $\sqrt[\dim U]{U}$, $U^T$, and $U^\dagger$ algorithms. Our results strengthen an advantage of linear optics, challenge the experiments on relaxed causality, and motivate new algorithms with many-outcome measurements.

quant-ph↗