The post Generic stabilizer in Spin14 appeared first on Skip Garibaldi.

]]>It is easier to see what is happening in SO_{14}. There one can see directly that the image of \(G_2 \times G_2\) has normalizer \((G_2 \times G_2) \rtimes \mu_4\), see p.62 of my 2009 Memoir of the AMS “Cohomological invariants” or in Markus Rost’s note “On 14-dimension quadratic forms, their spinors, and the difference of two octonion algebras“. In the \(\mu_4\) component of the normalizer of \(G_2 \times G_2\), you find that the element of order 2 is the linear transformation -1, which is in the center. (The inverse image of -1 in Spin_{14} is a 4th root of unity generating the center, which acts on the half-spin representation as a 4th root of unity and so does not fix a generic vector.)

Therefore, the normalizer of \(G_2 \times G_2\) in Spin_{14} is \((G_2 \times G_2) \rtimes \mu_8\), where a generator of the \(\mu_8\) interchanges the two \(G_2\)’s and is a square root of a generator of the central \(\mu_4\) in Spin_{14}. (This element is written explicitly in my Memoir.) For the earlier paper’s claim to be true, the element of order 2 in the \(\mu_8\) — a 4th power of a generator of the \(\mu_8\) — would have to interchange the two copies of \(G_2\). But its image in SO_{14} is 1, which doesn’t interchange the two components. Therefore, the stabilizer of a generic vector in the half-spin representation is \(G_2 \times G_2\) and no larger.

I am not sure what went wrong in the earlier paper. Perhaps the author found an element that lies not in Spin but in some slightly larger group. For example, if you consider \(G_2 \times G_2\) in SO_{14}, there is an element of O_{14} of order 2 (namely, the matrix that interchanges the two components and multiplies by -1, see the first paragraph on p.547 of my 2017 paper on Spin groups) that normalizes \(G_2 \times G_2\).

The post Generic stabilizer in Spin14 appeared first on Skip Garibaldi.

]]>The post Generically free representations appeared first on Skip Garibaldi.

]]>Back in the 1970s and 1980s, a bunch of people proved a host of results about finite-dimensional irreducible representations of simple Lie algebras over the complex numbers. They discovered a startling dichotomy: in various ways, small representations behave similarly to each other and large representations behave similarly to each other. Moreover, the difference between “small” and “large”, while the precise value depends on the algebra under consideration, is quite close to the dimension of the algebra.

Proving analogous results over fields of prime characteristic presents serious difficulties. For example, general techniques in characteristic zero prove the existence of a stabilizer in general position (i.e., an open set of vectors whose stabilizers are all conjugate), but this technique does not work in prime characteristic. A first step in transferring the bulk of the results to prime characteristic appears to be determining the generic stabilizer in each case. Moreover, determining the generic stabilizer has various other applications, such as to essential dimension.

In a series of papers with Bob Guralnick (part I, II, III and our earlier paper on spin groups) we settle this on the level of Lie algebras. Combined with a long work of Guralnick, Lawther, and Liebeck that determines the generic stabilizer on the level of groups of points, this determines the generic stabilizer as a group scheme.

I am ecstatic to be releasing these papers, as they have been a long time in the making. My part of this began shortly after I started work at IPAM at UCLA, back in fall of 2013.

The post Generically free representations appeared first on Skip Garibaldi.

]]>The post Nationwide lottery investigation by PennLive, Columbia School of Journalism appeared first on Skip Garibaldi.

]]>The reporters did a lot of legwork; this was a big effort. They got some mathematical assistance with the analysis from Philip Stark from UC Berkeley. The underlying math was developed in the paper Philip and I wrote with Richard Arratia from USC and Lawrence Mower from the Palm Beach Post, and the calculations can be done using this jupyter notebook from github.

The post Nationwide lottery investigation by PennLive, Columbia School of Journalism appeared first on Skip Garibaldi.

]]>The post Constructing a semisimple Lie algebra from its root system appeared first on Skip Garibaldi.

]]>He explains that it is a retelling of results on quantum groups from various works by George Lusztig, made simpler by translating them into the context of semisimple Lie algebras.

The construction goes by defining a vector space *M* with basis corresponding to a Chevalley basis of the Lie algebra — in this language a copy of the root system and a copy of the set of simple roots — and specific elements of *gl(M),* namely one element for each of the root subalgebras corresponding to plus or minus a simple root. He shows that the Lie subalgebra of *gl(M) *generated by these elements is semisimple with root system the one you started with.

The post Constructing a semisimple Lie algebra from its root system appeared first on Skip Garibaldi.

]]>The post Writing math papers appeared first on Skip Garibaldi.

]]>(Real beginners at writing math papers make a standard suite of mistakes, most of which have clear analogies with the problems described in Mark Twain’s Fenimore Cooper’s Literary Offences.)

The post Writing math papers appeared first on Skip Garibaldi.

]]>The post New York arrests two lottery “discounters” appeared first on Skip Garibaldi.

]]>The arrests come on the heels of an exposé by the New York Daily News on frequent lottery winners. Ticket discounting has been a hot topic for reporters since Lawrence Mower’s exposé of the activity in Florida for the Palm Beach Post that led to arrests and stores being banned from doing business with the lottery and contributed to the resignation of the Florida Lottery Secretary.

The mathematics behind Mower’s investigation is explained in this paper and this Jupyter notebook. Interestingly, claiming many big prizes is not enough to make someone a suspicious winner. The calculations described in the paper were able to distinguish ticket discounters and problem gamblers in Florida based only on publicly-available data about winners.

The post New York arrests two lottery “discounters” appeared first on Skip Garibaldi.

]]>The post Letter from E8 appeared first on Skip Garibaldi.

]]>*Gentlemen,*

*Mathematicians may be divided into two classes; those who know and love Lie groups, and those who do not. Among the latter, one may observe and regret the prevalence of the following opinions concerning the compact exceptional simple Lie group of rank 8 and dimension 248, commonly claled E _{8}.*

*That he is remote and unapproachable, so that those who desire to make his acquaintance are well advised to undertake an arduous course of preparation with E*_{6}and E_{7}.*That he is secretive; so that any useful fact about him is to be found, if at all, only at the end of a long, dark tunnel.**That he holds world records for torsion.*

*Point (1) deserves the following comment. Any right-thinking mathematician who wishes to construct the root-system of E _{6} does so as follows: first he constructs the root-system of E_{8}, and then inside it he locates the root-system of E_{6}. In this way he benefits from the great symmetry of the root-system of E_{8}, and its perspicuous nature. If this good precedent is not followed in their researches, one should consider whether to infer a lack of boldness in the investigator rather than a lack of cooperation from the subject-matter.*

*Since point (2) is equivalent ot point (1), we may pass to point (3). And here we should first reject the defences offered by some who might otherwise pass as well-informed. For they appear to regard it as a venial blemish on an otherwise worthy character, comparable to holding world records for the drinking of beer. This will not do. Let us first consider the rioutous profusion of torsion displayed by such groups as PSU(n). It then becomes clear that one can award a title to E _{8} only be restricting the competition to simply-connected groups. This is as if one where to award a title for drinking beer, having first fixed the rules so as to exclude all citizens of Heidelberg, Munich, Burton-on-Trent, and any other place where they actually brew or drink much of the stufff. In other words, it is contrary to natural justice.*

*In the second place, to consider the question at all reveals a certain preoccupation wtih ordinary cohomology. Any impartial observer must marvel at your obsession with this obscure and unhelpful invariant. The author, like all respectable Lie groups, is much concerned to present a decorous and seemly appearance to the eyes of K-theory; and taken in conjunction with other general theorems, this forces him to have a modest amount of torsion in ordinary cohomology. I shall seek some suitable person to inform you in an Appendix.*

*As a further argument against points (1) and (2), it is natural to release some small scrap of information which you would not otherwise possess. And this may also serve to guarantee the authenticity of this letter; for you must at least believe that it comes via some mathematician who would not mislead you about my views. You may then be expecting me to reveal, for example, H ^{*}(BE_{8}; F_{5}). I shall not oblige you. That could only encourage you in the low tastes that I have already condemned. Instead, I shall note the following possibility. It may happen that a space Y has its K-theory K^{*}(Y) torsion-free and zero in odd degrees, but nevertheless a careful study of K^{*}(Y) will reveal that Y must have torsion in its ordinary cohomology. Again, I shall seek some suitable person to inform you in an Appendix.*

*Be it therefore known and proclaimed among you, that my K-theory K(E _{8}) and that of my classifying space K(BE_{8}) cannot be criticised in this respect, at least at the prime 5. Their conduct is such as would be blameless and above reproach in the K-theory of a space without 5-torsion in its ordinary cohomology.*

*Given at our palace, etc, etc,*

* and signed*

* E _{8}.*

The post Letter from E8 appeared first on Skip Garibaldi.

]]>The post Moody lecture at Harvey Mudd appeared first on Skip Garibaldi.

]]>On Thursday, March 30th, come to Harvey Mudd for my talk “Identifying Lottery Scams Using Mathematics and Public Lottery Data”. This is the 2017 Michael E. Moody Lecture. Like other lectures in the series, it is aimed at a general audience.

This talk will tell the story of how a journalist, two mathematicians, and a statistician teamed up and used mathematics to identify people who were using the lottery as an adjunct to their illicit activities. The analysis combined old and new mathematics with on-the-ground detective work. The resulting series of journal and newspaper articles led to arrests and changes in state policy, and contributed to the resignation of the head of the Florida lottery. Still, many questions remain to be investigated.

The post Moody lecture at Harvey Mudd appeared first on Skip Garibaldi.

]]>The post Essential dimension article named Editor’s Choice appeared first on Skip Garibaldi.

]]>Here’s the abstract: 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 *dim G – 2(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 as 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.

The post Essential dimension article named Editor’s Choice appeared first on Skip Garibaldi.

]]>The post E8 survey to appear in Bulletin of the AMS appeared first on Skip Garibaldi.

]]>The post E8 survey to appear in Bulletin of the AMS appeared first on Skip Garibaldi.

]]>