arXiv subjects

Skip Garibaldi

Publications and source records attributed to Skip Garibaldi.

At least 19 recordsLinked to original sources

arXiv2026

The longest and shortest roots of a real cubic

There are many formulas in the literature providing roots of a real cubic that avoid some of the well-known pathologies of Cardano's formulas. Among these, we identify two that consistently provide the unique roots of a depressed cubic that have the greatest and smallest absolute value, whenever those exist. We call these the longest and shortest roots. The existence conditions are elementary and are in terms of the signs of the coefficients and the discriminant. Our proofs use two algebraic identities satisfied by hypergeometric functions; once the standard real branches are fixed, the root comparisons are entirely real. As an application, the longest-root formula gives an explicit factorization of all but a vanishing proportion of depressed real quartics.

math.CA↗

arXiv2026

Root of the generic cubic as a power series in the discriminant

An observation of J-P. Serre implies that the generic monic cubic polynomial, unique among generic monic polynomials of degree at least two, has a root that is a power series in the discriminant; Serre asked for a formula. We give one that works over any field with an absolute value and in every characteristic. Over a complete non-archimedean field of residue characteristic different from 3 we identify the root intrinsically: it is the isolated root, the one farthest from the others. We also answer the next case of Serre's question, computing explicitly the distinguished ramified quadratic factor of the generic monic quartic. The methods combine Hensel's lemma, Lagrange inversion, and elementary non-archimedean analysis.

math.RA↗

arXiv2025

Finding good bets in the lottery, and why you shouldn't take them

We give a criterion under which the expected return on a ticket for certain large lotteries is positive. In this circumstance, we use elementary portfolio analysis to show that an optimal investment strategy includes a very small allocation for such tickets.

math.HO↗

arXiv2024

Solutions to the exercises from the book "Albert algebras over commutative rings"

This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024, as well as errata and addenda. The addenda include proofs, in the style of the book, showing that (A1) Albert algebras are exceptional and in particular that a central simple Jordan algebra over a field is exceptional if and only if it is an Albert algebra; (A2) A regular lattice in a real Albert algebra is also an Albert algebra; (A3) a Freudenthal algebra over a field is split by an extension of degree dividing 6; and (A4) a Freudenthal subalgebra of rank 9 in an Albert algebra can be used to describe the Albert algebra as a Tits construction.

math.RA↗

arXiv2023

Invariant derivations and trace bounds

About 20 years ago, J-P.~Serre announced a bound on the trace of elements of compact Lie groups under the adjoint representation together with related results, provided indications of his proofs, and invited a better proof. This note provides a new, general method for proving such bounds; uses that method to derive Serre's bounds; gives a second proof of Serre's announced results that (we learned) closely follows his original argument; and provides lower bounds for traces of other representations of compact Lie groups and for Brauer characters of finite groups.

math.RT↗

arXiv2022

Pictures of compact Lie groups (after Serre)

We fill in the details in a procedure outlined by Serre for drawing pictures of compact real Lie groups. In the case of Sp($2n$), the picture generated by the method is connected with abelian varieties over a number field or a finite field. We follow the procedure to produce pictures for the three simply connected simple groups of rank 2. The pictures for two of these have previously been discussed in the literature in a different setting. The remaining one, type $G_2$, has the most complicated picture.

math.GR↗

arXiv2022

Albert algebras over Z and other rings

Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative non-associative algebras and also arise naturally in the context of simple affine group schemes of type $F_4$, $E_6$, or $E_7$. We study these objects over an arbitrary base ring $R$, with particular attention to the case of the integers. We prove in this generality results previously in the literature in the special case where $R$ is a field of characteristic different from 2 and 3.

math.RA↗

arXiv2021

Generic stabilizers for simple algebraic groups

We prove a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of $G$ and the group $G(k)$ of $k$-points. For $G$ simple and $V$ faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those $G$ and $V$ for which the stabilizer in general position is smooth, or $\dim V/G < \dim G$, or there is a $v \in V$ whose stabilizer in $G$ is trivial.

math.RT↗

arXiv2020

Minuscule embeddings

We study embeddings $J \rightarrow G$ of simple linear algebraic groups with the following property: the simple components of the $J$ module Lie($G$)/Lie($J$) are all minuscule representations of $J$. One family of examples occurs when the group $G$ has roots of two different lengths and $J$ is the subgroup generated by the long roots. We classify all such embeddings when $J = SL_2$ and $J = SL_3$, show how each embedding implies the existence of exceptional algebraic structures on the graded components of Lie($G$), and relate properties of those structures to the existence of various twisted forms of $G$ with certain relative root systems.

math.RT↗

arXiv2020

A class of continuous non-associative algebras arising from algebraic groups including $E_8$

We give a construction that takes a simple linear algebraic group $G$ over a field and produces a commutative, unital, and simple non-associative algebra $A$ over that field. Two attractions of this construction are that (1) when $G$ has type $E_8$, the algebra $A$ is obtained by adjoining a unit to the 3875-dimensional representation and (2) it is effective, in that the product operation on $A$ can be implemented on a computer. A description of the algebra in the $E_8$ case has been requested for some time, and interest has been increased by the recent proof that $E_8$ is the full automorphism group of that algebra. The algebras obtained by our construction have an unusual Peirce spectrum.

math.RA↗

arXiv2018

Generically free representations III: extremely bad characteristic

In parts I and II, we determined which faithful irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for Lie($G$), i.e., which $V$ have an open subset consisting of vectors whose stabilizer in Lie($G$) is zero, with some assumptions on the characteristic of the field. This paper settles the remaining cases, which are of a different nature because Lie($G$) has a more complicated structure and there need not exist general dimension bounds of the sort that exist in good characteristic.

math.RT↗

arXiv2017

Generically free representations II: irreducible representations

We determine which faithful irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for Lie($G$), i.e., which $V$ have an open subset consisting of vectors whose stabilizer in Lie($G$) is zero. This relies on bounds on $\dim V$ obtained in prior work (part I), which reduce the problem to a finite number of possibilities for $G$ and highest weights for $V$, but still infinitely many characteristics. The remaining cases are handled individually, some by computer calculation. These results were previously known for fields of characteristic zero, although new phenomena appear in prime characteristic; we provide a shorter proof that gives the result with very mild hypotheses on the characteristic. (The few characteristics not treated here are settled in part III.) These results are related to questions about invariants and the existence of a stabilizer in general position.

math.RT↗

arXiv2017

Generically free representations I: large representations

For a simple linear algebraic group $G$ acting faithfully on a vector space $V$ and under mild assumptions, we show: if $V$ is large enough, then the Lie algebra of $G$ acts generically freely on $V$. That is, the stabilizer in the Lie algebra of $G$ of a generic vector in $V$ is zero. The bound on $\dim V$ grows like $(\mathrm{rank} G)^2$ and holds with only mild hypotheses on the characteristic of the underlying field. The proof relies on results on generation of Lie algebras by conjugates of an element that may be of independent interest. We use the bound in subsequent works to determine which irreducible faithful representations are generically free, with no hypothesis on the characteristic of the field. This in turn has applications to the question of which representations have a stabilizer in general position as well as the determination of the invariants of the representation.

math.RT↗

arXiv2016

Globally Irreducible Weyl Modules for Quantum Groups

The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $E_{8}$ or its highest weight is minuscule. In this paper, we prove an analogous criteria for irreducibility of Weyl modules over the quantum group $U_{\zeta}({\mathfrak g})$ where ${\mathfrak g}$ is a complex simple Lie algebra and $\zeta$ ranges over roots of unity.

math.RT↗

arXiv2016

Globally Irreducible Weyl Modules

In the representation theory of split reductive algebraic groups, it is well known that every Weyl module with minuscule highest weight is irreducible over every field. Also, the adjoint representation of $E_8$ is also irreducible over every field. In this paper, we prove a converse to these statements, as conjectured by Gross: if a Weyl module is irreducible over every field, it must be either one of these, or trivially constructed from one of these.

math.RT↗

arXiv2016

Spinors and essential dimension

We prove that spin groups act generically freely on various spinor modules, in the sense of group schemes and in a way that does not depend on the characteristic of the base field. As a consequence, we extend the surprising calculation of the essential dimension of spin groups and half-spin groups in characteristic zero by Brosnan--Reichstein--Vistoli (Annals of Math., 2010) and Chernousov--Merkurjev (Algebra & Number Theory, 2014) to fields of characteristic different from 2.

math.GR↗

arXiv2015

Essential dimension of algebraic groups, including bad characteristic

We give upper bounds on the essential dimension of (quasi-)simple algebraic groups over an algebraically closed field that hold in all characteristics. The results depend on showing that certain representations are generically free. In particular, aside from the cases of spin and half-spin groups, we prove that the essential dimension of a simple algebraic group $G$ of rank at least two is at most $\mathrm{dim}(G) - 2(\mathrm{rank}(G)) - 1$. It is known that the essential dimension of spin and half-spin groups grows exponentially in the rank. In most cases, our bounds are as good or better than those known in characteristic zero and the proofs are shorter. We also compute the generic stabilizer of an adjoint group on its Lie algebra.

math.GR↗

arXiv2015

On the Tits p-indexes of semisimple algebraic groups

The first author has recently shown that semisimple algebraic groups are classified up to motivic equivalence by the local versions of the classical Tits indexes over field extensions, known as Tits p-indexes. We provide in this article the complete description of the values of the Tits p-indexes over fields. From this exhaustive study, we also deduce criteria of motivic equivalence for semisimple groups of many types, hence giving a dictionary between classic algebraic structures, representation theory, cohomological invariants and Chow motives of the twisted flag varieties for those groups.

math.AG↗