## Groups, Nilpotence, and Tensors

A regional workshop featuring research on groups, nilpotent structures, and their associated multilinear algebra.

**Zoom Link Please E-mail James.Wilson@ColoState.Edu**

## Locations

- Friday Social: Prost contact Peter Brooksbank (pbrooksb@bucknell.edu) for details.
- Talks & Coffee: Weber Building, Colorado State University, Room 223.
- Dinner: Sherpa Grill 2 Fort Collins

## Schedule

Talks will take place on Saturday April 29 beginning at **9am** (Coffee and snacks to be served starting at **8:30am**).

Time |
Event |
---|---|

8:30 - 9:00 | Tea/Coffee |

9:00 - 9:25 | Peter Brooksbank, Computing intersections of classical groups |

9:30 - 9:55 | Mandi A. Schaeffer Fry, Recent Results on Local-Global Conjectures |

10:00 - 10:30 | Break/Tea |

10:30 - 10:55 | Aparna Upadhyay, Some conjectures on tensor products of modular representations |

11:00 - 11:25 | Alex Ryba, Relative eigenvector and pure tensor problems |

11:30 - 11:55 | Chris Liu, Quick solvers for endomorphisms of modules |

12:00 - 1:30 | Lunch (not provided) |

1:30 - 1:55 | Tobias Rossmann, Toric Tools in Asymptotic Group Theory |

2:00 - 2:25 | Martin Kassabov, Subgroups of finite simple groups are maximally diverse |

2:30 - 2:55 | Angela Carnevale, Growth in nilpotent Lie rings |

3:00 - 4:00 | Break/Tea |

4:00 - 4:25 | Michael Levet, On the Descriptive Complexity of Finite Groups |

4:30 - 4:55 | Justin Lynd, Realizing finite groups as automizers |

5:00 - 5:25 | Petr Vojtechovsky, Nilpotence just beyond groups |

**Dinner at Sherpa Grill 2 Fort Collins** 6PM! The location is 1 mile from where we are having the conference we can walk over together.

But in the tradition of Southwest Group Theory day, talks are planned throughout Saturday April 29 with the hope that collaborations and discussions can spill over to Sunday for those who can’t usually meet together and use this as an excuse to get together again.

## Participants

Thanks to all the in-person and on-line participants for a day of learning and fun!

- Page Wilson (Colorado State U.)
- James B. Wilson (Colorado State U.)
- Sean Willmot (Colorado State U.)
- Petr Vojtechovsky (University of Denver)
- Aparna Upadhyay (U. Arizona)
- Nathaniel Thiem (U. Colorado)
- Kylie Schooner (Colorado State U.)
- Mandi Schaeffer Fry (Metro State U.)
- Tobias Rossmann (U. Galway)
- Alex Ryba (CUNY)
- Chris Parker (v) (U. Birmingham)
- Eric Moorhouse (U. Wyoming)
- Amaury Minino (Colorado State U.)
- Josh Maglione (v) (Otto von Guericke U.)
- Keith Kearnes (v) (U. Colorado)
- Bill Kantor (v) (U. Oregon)
- Delaram Kahrobaei (v) (CUNY)
- Ian Jorquera (Colorado State U.)
- Justin Lynd (U. Louisiana at Lafayette)
- Michael Levet (U. Colorado)
- Chris Liu (Colorado State U.)
- Martin Kassabov (Cornell U.)
- Alexander Hulpke (Colorado State U.)
- Yassine Guerboussa (v) (U. Kasdi Merbah Ouargla)
- Angela Carnevale (U. Galway)
- Pete Brooksbank (Bucknell U.)

### Registration

Day-of, get a name tag, but no cost. Sponsored by Dept. Math. Colorado State University.

## Talks

We will make rooms available on Saturday and Sunday for collaborations.

- Petr Vojtechovsky, Nilpotence just beyond groups Slides
- Aparna Upadhyay, Some conjectures on tensor products of modular representations.
- Mandi A. Schaeffer Fry, Recent Results on Local-Global Conjectures
- Tobias Rossmann, Toric Tools in Asymptotic Group Theory
- Alex Ryba, Relative eigenvector and pure tensor problems Slides
- Justin Lynd, Realizing finite groups as automizers
- Michael Levet, On the Descriptive Complexity of Finite Groups Slides
- Chris Liu, Quick solvers for endomorphisms of modules. Slides
- Martin Kassabov, Subgroups of finite simple groups are maximally diverse
- Angela Carnevale, Growth in nilpotent Lie rings
- Peter Brooksbank, Computing intersections of classical groups

## Sponsors

- Colorado State University, Department of Mathematics

### Organizers

- Alexander Hulpke, Colorado State U. Dept. Mathematics.
- James B. Wilson, Colorado State U. Dept. Mathematics.

#### Petr Vojtechovsky

**Title:** Nilpotence just beyond groups

**Abstract:** The concepts of nilpotence, supernilpotence and solvability can be defined quite generally in congruence modular varieties. While in groups nilpotence and supernilpotence coincide, they differ in varieties very close to groups. Even in groups it might be instructive to see why the two rather different concepts happen to coincide. If time permits, we will also discuss the “correct” definition of solvability and why again it happens to coincide with the standard notion of solvability from groups. I will present some open problems.

#### Aparna Upadhyay

**Title:** Some conjectures on tensor products of modular representations.

**Abstract:** Examining the rate of growth of the non-projective part of tensor powers of a modular representation of a finite group led to some interesting conjectures by Dave Benson and Peter Symonds. Some very basic questions about the decompositions of tensor products arise. I will discuss the recent progress on some of the conjectures.

#### Mandi A. Schaeffer Fry

**Title:** Recent Results on Local-Global Conjectures

**Abstract:** The local-global conjectures in character theory seek to relate the character theory of a finite group to the character theory and properties of certain nice, “local” subgroups. I will discuss some of these conjectures and recent progress.

#### Tobias Rossmann

**Title:** Toric tools in asymptotic group theory

**Abstract:** Various enumerative problems surrounding nilpotent groups can be reduced to (often difficult) geometric problems pertaining to tensors. I will describe some cases where such geometric challenges can be tackled using ideas and tools from toric geometry.

#### Alex Ryba

**Title:** Relative eigenvector and pure tensor problems

**Abstract:** If you think those would be of interest. (I do!)

#### Martin Kassabov

**Title:** Subgroups of finite simple groups are maximally diverse

**Abstract:** Give me till the end of the day.

#### Justin Lynd

**Title:** Realizing finite groups as automizers

**Abstract:** Motivated by a related question of P. Mueller, we wondered whether each finite group A could be realized as an automizer $\mathsf{Aut}_G(U)$ in some finite group $G$ with subgroup $U$. This is easy to do if $U$ is allowed to be normal in $G$, for then one can take a faithful representation of $A$ on an elementary abelian $p$-group $U$ for some prime $p$ and form the semidirect product $G = UA$. I will discuss joint work with Sylvia Bayard in which we are able to realize $A$ as an automizer $\mathsf{Aut}_G(U)$ even under the requirement that the $G$-conjugates of $U$ generate $G$. The method of construction is somewhat curious in that we do not know how to do it without using some initial pieces of the theory of fusion systems on finite groups.

#### Chris Liu

**Title:** Quick solvers for endomorphisms of modules.

**Abstract:** Computing the endomorphism ring of a module over an algebra and the adjoint algebra of a bilinear map both require solving systems matrix equations of the form $XA_i + B_iY = C_i$. Given $A$, $B$, $C$ length $n$ lists of $n$-by-$n$ matrices we face a system of $n^3$ unknowns in $2n^2$ equations. This talk outlines an $O(n^3)$ solution in the general case. The key is to view $A,B,C$ as 3-tensors and apply invertible linear transformations that can be computed piecewise, in analogy to solving a linear system $Mx = b$ by left-multiplying by $M^{-1}$.

#### Michael Levet

**Title:** On the Descriptive Complexity of Finite Groups

**Abstract:** In this talk, I will introduce the first and second Ehrenfeucht–$Fra"{ï}ss'e$ pebble game in Hella’s hierarchy, which provide combinatorial measures of complexity for relational structures. There is a classical correspondence between this first pebble game, the logic $\textsf{FO} + \textsf{C}$ (first-order logic with counting quantifiers), and the Weisfeiler–Leman algorithm. As a consequence, this first pebble game allows us to make simultaneous advances regarding both logical definability (descriptive complexity) and the parallel (circuit) complexity of isomorphism testing. The second Ehrenfeucht–$Fra"{ï}ss'e$ pebble game in Hella’s hierarchy corresponds to the logic $\textsf{FO}(Q)$, where $Q$ is the set of all generalized binary quantifiers. While this second game trivially solves graph isomorphism, it does not (obviously) do so in the setting of groups and other ternary relational structures.

One goal of this talk will be to learn how to reason about finite groups using these pebble games. I will also discuss recent work with Joshua A. Grochow, in which we leverage this second pebble game to show that Fitting-free groups are identified by formulas in $\textsf{FO}(Q)$ that use only $O(1)$ variables and $O(1)$ quantifier depth.

#### Angela Carnevale

**Title** Growth in nilpotent Lie rings

**Abstract** I will talk about recent progress in the study of ideal zeta functions of class-2 nilpotent Lie rings. In particular, I’ll discuss how a new family of combinatorially-defined rational functions, so-called generalised Igusa functions, allows us to compute formulae for the ideal zeta functions of large families of rings. This is based on joint work with Michael Schein and Christopher Voll.

#### Peter Brooksbank

**Title:** Computing intersections of classical groups

**Abstract:** Largely due to its applications to p-groups, understanding the intersection of a set of classical groups has long been a problem of interest in computational algebra. It reduces to the following computational problem: given a set of reflexive forms on a common vector space, construct generators for the group of invertible linear transformations that preserve every form in the set.

The first effective algorithms to solve the problem were developed for specific pairs of forms in joint work with E.A. O’Brien as long ago as 2008. More generally, by viewing the individual forms instead as a single bilinear map, the target group of isometries can be realized as the group of unitarian elements of an algebra with involution. This perspective was exploited in a joint 2012 paper with J.B. Wilson to develop a polynomial-time algorithm to solve the isometry group problem over finite fields of odd characteristic.

Unfortunately, the approach taken in the 2012 paper founders over fields of characteristic 2 in two significant ways, and despite continued interest in the problem this case has remained open for the past decade. In this talk I will present some recent developments with the characteristic 2 case. This is a report on joint work with M. Kassabov and J.B. Wilson.