User michael joyce - MathOverflow

Answer by Michael Joyce for What are the most important open problems in algebraic combinatorics?

2013-04-22T16:25:51Z

One basic problem is to give a combinatorial formula (or "Littlewood-Richardson rule") for the Schubert structure constants. There is a basis $\{ X_w \}$ of the polynomial ring $\mathbb{Z}[x_1,x_2,\dots]$ in countably many variables that is indexed by $w \in S_{\infty}$, where $S_{\infty}$ is the group of permutations of $\mathbb{N} = \{1,2,\dots\}$ that fix all but finitely many numbers. Given $u, v \in S_{\infty}$, one can expand the product $X_u X_v$ in this basis:

$$ X_u X_v = \sum_{w \in S_{\infty}} c_{u,v}^w X_w. $$

A priori, one only knows that $c_{u,v}^w \in \mathbb{Z}$, but it is known for geometric reasons that in fact $c_{u,v}^w \geq 0$ for all $u,v,w \in S_{\infty}$. The problem is to find a positive formula for $c_{u,v}^w$; loosely speaking, the goal is to find a set of objects that is counted by $c_{u,v}^w$.

The problem is intimately tied up with the geometry of the (complete) flag varieties, just as the Littlewood-Richardson rule is connected to the geometry of the Grassmannians.

Also, everything can be formulated for polynomials in finitely many variables indexed by a finite symmetric group $S_n$, but in that case the statement is really about a quotient of a polynomial ring. There is also a generalization to Weyl groups of other types if work with a quotient of a polynomial ring (the Borel presentation of the cohomology of a flag variety).

Edit: I'm pretty sure this problem is mentioned in Stanley, but it is worth emphasizing that it is still open and that there is a lot of beautiful combinatorics related to it. Recently, progress on the two-step flag variety version of the problem was made due to conjectures of Allen Knutson and Ravi Vakil. Izzet Coskun proved A Littlewood-Richardson rule for two-step flag varieties using the notion of Mondrian tableaux. Anders Buch has conjectured a rule for three-step flag varieties.

Seeking a generalization of group embedding of symmetric varieties

2013-04-16T19:59:19Z

I am looking for generalizations of the following construction.

Let $G$ be a connected, reductive group and let $\theta : G \rightarrow G$ be an involution. Let $H = G^{\theta}$ be the subgroup of $\theta$-fixed points. Then $G/H$ is known as a symmetric variety. The map $\tau: G \rightarrow G$ given by $\tau(g) = g \theta(g)^{-1}$ descends to an embedding $\overline{\tau} : G / H \hookrightarrow G$.

I would like to consider a more general setting. Let $G$ be a connected, reductive group and $H$ a spherical subgroup of $G$ (meaning that some Borel subgroup of $G$ has a dense orbit in $G/H$ under the usual left multiplication action). Let $G'$ be another reductive group. I am interested in examples of the following:

a morphism of algebraic groups $\alpha : G \rightarrow G' \times G'$; let $\alpha_1, \alpha_2 : G \rightarrow G'$ be the corresponding morphisms obtained by composing with the two projections;
a locally closed $\alpha$-equivariant embedding $f : G / H \hookrightarrow G'$, where $\alpha$-equivariance means that $f(g \cdot xH) = \alpha_1(g) f(xH) \alpha_2(g)^{-1}$.

The embedding of a symmetric variety $G/H$ into $G$ fits into this setup by taking $G' = G$ and $\alpha(g) = (g, \theta(g))$. I would like to ask if anyone knows of any other examples of this general construction in the literature, or results that would rule out such constructions in certain situations. My expectation is that, if such a construction exists for a wider class of spherical varieties, then the group $G'$ will likely be considerably larger than $G$.

Edit: The construction for symmetric varieties is used in work of R.W. Richardson and T.A. Springer (The Bruhat order on symmetric varieties and Combinatorics and geometry of K-orbits on the flag manifold). The generalized construction outlined above would possibly aid in a study of Bruhat order on a wider class of spherical varieties.

Answer by Michael Joyce for Partial order relation on subsets

2013-02-14T15:52:14Z

This is the Bruhat order on $S_n / (S_k \times S_{n-k})$, which models the inclusion relations of Schubert varieties on the Grassmannian $Gr(k,n)$. The elements of $S_n / (S_k \times S_{n-k})$ are generally modeled by their minimal length representatives, called Grassmann permutations. You can biject your $k$-subset of $[n]$ to a Grassmann permutation by sending a $k$-subset $\lbrace{ a_1, \dots, a_k \rbrace}$ (with $a_1 < a_2 < \dots < a_k$) to the permutation of $[n]$ whose window (one-line) notation is $a_1 a_2 \cdots a_k b_1 b_2 \cdots b_{n-k}$ with $\lbrace{ b_1, b_2, \dots, b_{n-k} \rbrace}$ denoting the complement of your given set, again ordered in increasing order $b_1 < b_2 < \cdots < b_{n-k}$.

Answer by Michael Joyce for Trichotomies in mathematics

2013-02-03T00:52:53Z

There are three types of subgroups of $PGL_2(\mathbb{C})$ that act on $\mathbb{P}^1$ non-transitively but with finitely many orbits:

(1) Type $T$: a one-dimensional torus

(2) Type $N$: the normalizer of a one-dimensional torus

(3) Type $U$: containing a non-trivial one-dimensional unipotent subgroup

This trichotomy plays a key role in the study of the geometry of spherical varieties, a class of algebraic varieties that includes grassmannians, flag varieties, toric varieties, algebraic monoids and symmetric spaces. It is particularly important in understanding the analogues of Schubert subvarieties (i.e. closures of orbits of a Borel subgroup) of a spherical variety.

In this example, there is no "middle" case as there is no intrinsic order to the three types.

Answer by Michael Joyce for Graph of $S_n$ with respect to transposition

2012-12-29T21:43:14Z

This is the undirected version of the Bruhat graph. To make the graph directed, direct an edge $\sigma \rightarrow \sigma'$ if $\ell(\sigma') > \ell(\sigma)$, where $\ell(\sigma)$ denotes the length of $\sigma$ defined to be the number of inversions of $\sigma$. A related graph is the Hasse diagram of the Bruhat order, which is the subgraph of the Bruhat graph where only the edges $\sigma \rightarrow \sigma'$ with $\ell(\sigma') = \ell(\sigma) + 1$ are kept. A basic fact is that for any edge $\sigma \rightarrow \sigma'$ in the Bruhat graph with $\ell(\sigma') - \ell(\sigma) > 1$, there is a path of edges in the Hasse diagram starting at $\sigma$ and ending at $\sigma'$.

A standard reference for this material is Chapter 2 of Bjorner and Brenti's Combinatorics of Coxeter Groups.

Answer by Michael Joyce for What would you want to see at the Museum of Mathematics?

2012-11-30T22:06:37Z

I'd suggest an interactive exhibit where people can tweak the parameters of a population model with 3 species in it. Have an information panel which explains what the parameters represent. Suggest goals such as (1) keep the rabbits from going extinct, (2) create a stable equilibrium where all three species survive, (3) find a cyclic solution, etc. Experts could probably come up with a good system that exhibited lots of interesting behavior. Let people visualize their solutions both graphically (3-D graph of all three populations as well as 2-D graphs of any two populations of their choosing). There are probably other clever graphical representations that others could come up with as well.

Overall, I think this is a great endeavor and I hope that you will focus on making the exhibits interactive, with pathways to learning. Ideally, the same person could visit an exhibit a half dozen times and learn something new each time. Of course, making things fun and interesting is really important too - but I think that should naturally emerge from the design of interactive ways to explore a beautiful piece of mathematics.

Answer by Michael Joyce for Examples where adding complexity made a problem simpler

2012-11-25T03:52:24Z

Much of modern algebraic geometry fits this paradigm. The introduction of schemes makes some things much more complicated -- non-uniqueness of the embedded components of the primary decomposition compared to the uniqueness of the decomposition of a variety into its irreducible components comes to mind -- but the technical advantages of scheme theory have proven themselves many times over in the 50+ years since they were introduced.

Answer by Michael Joyce for Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups

2012-11-07T13:37:52Z

(1) Assuming you are referring to the coefficient field for your cohomology theory, then yes, the result immediately extends to $\mathbb{R}$ and $\mathbb{C}$ coefficients.

(2) Two sources are Fulton's Young Tableaux (Chapters 9 and 10) and Manivel's Symmetric Functions, Schubert Polynomials, and Degeneracy Loci (Chapter 3). Both of these sources assume a basic knowledge of and familiarity with algebraic geometry.

Edit: The original source for the result is a paper of Armand Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57, (1953), 115–207.

Answer by Michael Joyce for Intersection theory for $G$-varieties - an action on the chow ring?

2012-10-17T20:05:08Z

If you are interested in intersection theory of varieties with $G$-actions, then you want to study equivariant intersection theory. This theory exploits the $G$-action in a way that leads to deeper invariants than ordinary intersection theory. The three references I would recommend if you are first learning the subject are:

Fulton's lectures notes on equivariant cohomology (compiled by Dave Anderson)

Equivariant Cohomology and Equivariant Intersection Theory by Michel Brion

Equivariant Chow Groups for Torus Actions by Michel Brion

Answer by Michael Joyce for References request on the algebraic geometry of projective homogeneous spaces

2012-10-11T19:18:27Z

Michel Brion's Lectures on the Geometry of Flag Varieties answers all of your questions in the special case $G = SL_n$ and $P = B$ (the Borel subgroup of upper triangular matrices). See Section 1.4. If you are not so familiar with this particular field, you may find the entire first section quite helpful.

Answer by Michael Joyce for Continuous notions with compelling discrete analogues

2012-08-20T04:06:19Z

Discrete difference equations generalize differential equations. In a similar spirit, divided difference operators generalize partial differentiation operators. Though such operators go back to Newton, there has a been a resurgence of interest in them since the work of Lascoux and Schutzenberger on Schubert polynomials. While partial differentiation operators satisfy commutativity relations $\partial_x \partial_y = \partial_y \partial_x$, the divided difference operators satisfy the nilHecke relations. This gives the discrete operators a certain richness that is not present in the continuous operators.

Answer by Michael Joyce for Interesting mathematical documentaries

2012-06-19T19:59:09Z

Marcus du Sautoy's The Story of Maths is a total of four hours attempting to give an overview of the history of mathematics from ancient to modern time, spending 5-10 minutes each on the life and work of some of the most famous mathematicians. While one could quibble with some of the selections, the project is overall a fantastic production.

Answer by Michael Joyce for Where to publish a math textbook in Creative Commons

2012-06-19T14:58:24Z

You might try contacting Lon Mitchell at Virginia Commonwealth University. VCU has published hardcover versions of two open source textbooks (one on Linear Algebra, the other on Abstract Algebra), each costing under $20.

Answer by Michael Joyce for The pseudoeffective cone does not contain lines

2012-06-12T15:32:22Z

First, whether a class is pseudoeffective or not depends only on its numerical equivalence class. (The pseudoeffective cone is the closure of the cone of big classes, and $D$ is big if and only if $nD$ is numerically equivalent to $A + E$, where $A$ is an ample divisor on $X$, $E$ is an effective divisor on $X$ and $n > 0$ is an integer.)

Now suppose a class $\beta$ was such that both $\beta$ and $-\beta$ were pseudoeffective. Then $\beta \cdot a_1 \cdots a_k \geq 0$ and $\leq 0$ for any ample classes $a_1, \dots, a_k$. So $\beta \cdot a_1 \cdots a_k = 0$. But any divisor class can be written as a difference of ample classes, so $\beta \cdot \delta_1 \cdots \delta_k = 0$ for all divisor classes $\delta_1, \dots, \delta_k$. Thus, $\beta = 0$.

The standard reference for this material is Lazarsfeld's Positivity in Algebraic Geometry. I don't have a copy handy, but I'm $> 99\%$ certain that this fact is proved somewhere in Volume I.

Edit: I should add that this assumes that $X$ is projective. I don't know of any results without this assumption.

Answer by Michael Joyce for Good combinatorics textbooks for teaching undergraduates?

2012-05-17T15:51:27Z

Neither of these suggestions seem to exactly fit the level the OP was aiming for, but I add them for others who come across this thread with a different group of students in mind:

(1) For a gentle, problem-based introduction for undergraduates, I really like Ken Bogart's Combinatorics Through Guided Discovery. Sadly, he passed away while writing the text, but he has left it publically available at no charge.

(2) For a comprehensive and structured approach to combinatorics at the introductory graduate level, I really like Martin Aigner's A Course in Enumeration.

Is there a brief name for the symmetric space $SL_{2n} / Sp_{2n}$?

2012-05-16T11:02:48Z

Let $V$ be a complex vector space of even dimension. Then the homogeneous space $SL_{2n}(V) / SP_{2n}(V)$ is known to parametrize the space of non-degenerate skew-symmetric bilinear forms on $V$.

(1) Can anyone provide a reference in the literature which refers to this space with a name that is briefer than "the space of non-degenerate skew-symmetric bilinear forms on $V$"?

(2) If there seem to be no adequate answers to question (1) in the affirmative, does anyone have any suggestions for a compact name of this space?

Answer by Michael Joyce for Theory of cones

2012-03-24T16:46:38Z

If you are interested in the lattice point enumeration aspect, I'd also suggest Computing the Continuous Discretely by Beck and Robins. There's a version of the book available on their website which you can use to preview it.

Answer by Michael Joyce for Chains in $K\backslash G/B$ lying over a closed $K$-orbit

2012-03-22T00:21:13Z

Malheureusement, this is not true, not even for the weak order. This can be seen for example when $G = GL(4)$ and $K = GL(2) \times GL(2)$. Then $K \backslash G / B$ is parameterized by involutions with signs attached to fixed points and the map $\varphi$ simply forgets the markings on the fixed points.

For example, the closed orbit associated to $(1^+)(2^-)(3^-)(4^+)$ lies below both $(14)(2^-)(3^+)$ and $(14)(2^+)(3^-)$. Figures with weak order for a plethora of examples of symmetric subgroups appear in a preprint by Ben Wyser.

Answer by Michael Joyce for Covering relations in $K\backslash G/B$

2012-03-19T20:00:11Z

While the paper of Richardson-Springer does study the weak order, it also has useful results on the usual (strong) Bruhat order. In particular, Theorem 7.11 says that Bruhat order is characterized as the weakest partial order on $K \backslash G / B$ that contains the weak order and satsifies the obvious geometric condition $Y \subseteq Y' \Rightarrow P_s \cdot Y \subseteq P_s \cdot Y'$ where $Y$ and $Y'$ are arbitrary $B$-stable subvarieties and $P_s$ denotes the minimal parabolic subgroup associated to the simple reflection $s$.

I don't know a specific reference in the literature for the results you want, but I suspect they can be proven using the generalized reduced decomposition that is introduced in R-S.

Answer by Michael Joyce for Where does a math person go to learn quantum mechanics?

2012-01-15T03:49:32Z

Igor Dolgachev's course notes "Introduction to Physics" starts with an introduction to classical mechanics and develops quantum mechanics from a mathematical point of view. It's a good place to start if you're strong mathematically but are having a difficulty understanding physicists' notations and perspectives.

Answer by Michael Joyce for Learning Tropical geometry

2011-12-31T18:20:35Z

Several years ago, I participated in a learning seminar in tropical algebraic geometry and collected several helpful survey articles. (This was before Maclagan and Sturmfels' book was written, which I suspect is excellent.)

Anyway, here were some of the most helpful intro points for me: Tropical Mathematics, First Steps in Tropical Geometry, Tropical Algebraic Geometry, Introduction to Tropical Geometry, The Tropical Grassmannian, The Number of Tropical Plane Curves Through Points in General Position.

Sturmfels, Speyer, and Gathmann all write very well, and Gathmann especially devotes considerable space to giving motivation for the field. Mikhalkin, of course, was the one who pioneered the idea of attacking challenging classical problems (such as counting the number of plane curves of genus $g$ and degree $d$ passing through $3d + g - 1$ points, which had just been solved by Capraso-Harris in the late 90s) using the tropical semifield.

Answer by Michael Joyce for Motivating Algebra and Analysis for Average Undergraduates

2011-11-01T21:25:37Z

As far as abstract algebra goes, I think you should make the analogy with your students that when they learned the four basic operations as a child that they did not at the time have full appreciation of what they were learning. To explain to a seven year old the "application" of balancing a checkbook or calculating the tip on a check would have little meaning to them. Similarly, when learning about groups or rings for the first time, they are entering a whole new world of calculations that they can perform. At first, they will learn the rules without a full understanding of what they are doing. In due time (i.e. later in the course), after much hard work on their part, their understanding will have increased and they will be ready to appreaciate the applications of their new skills. I generally find students are willing to give you some slack, at least for a while, but you do need to make sure you fulfill your promise so that the students do get at least a glimpse of some interesting applications before the course is done.

As far as specific applications go, using Burnside's Theorem for enumeration problems is a fun application of finite groups that students enjoy once they "get" it. It is also worthwhile, assuming the students have a linear algebra course in their background, to go back and reinterpret diaganolization and Jordan normal form in terms of group actions. It is criminal how many math majors get a degree without having any concept of why diagonalization and Jordan normal form matter, despite how important eigenvalue-eigenvector analysis is in real world science and engineering applications. It should be possible to introduce the basic idea of Klein's Erlangen program (in the familiar example of Euclidean geometry) for students to see another example of symmetry at work, though this might be a bit ambitious. I would avoid problems such as classifying finite simple groups of order $< n$ (your choice of $n$). I think that the few students who actually are stimulated by such problems are the ones who generally need the most encouragement to unleash their creative side to complement their appreciation for logical rigor.

Answer by Michael Joyce for Dimension of spaces of invariants/tableaux functions

2011-10-20T10:18:06Z

The numbers you refer to are known as Kostka numbers. They are discussed in standard references like Fulton's Young Tableaux and Stanley's Enumerative Combinatorics. The weights of a tableaux are often referred to as their content, as well.

Answer by Michael Joyce for What is the "right" deifinition of the homology(cohomology) of an orbifold?

2011-10-05T16:08:44Z

I do not know the current state of the art, but I can point you to the paper of Chen and Ruan which should have everything you are asking for.

Answer by Michael Joyce for Configuration space of flags

2011-09-29T21:49:37Z

(This should be a comment but I don't have enough reputation to leave one.) The quotient of $U$ by $PGL(2)$ is $\mathbb{P}^1 \backslash { 0, 1, \infty }$. To get something compact, you need to allow the four points to lie on a stable curve -- in this case, you need to allow the union of two $\mathbb{P}^1$'s with two of your points on each component (none of the points coinciding with the node of intersection).

You might want to look at Multiple Flag Varieties of Finite Type by Magyar, Weyman, and Zelevinsky. It characterizes the conditions under which you get finitely many $PGL(m)$ orbits on products of (partial) flag varieties and may help give a sense of the complexities involved.

Answer by Michael Joyce for the blowing up of a plane curve playing me tricks.

2011-08-31T05:53:05Z

On your third blowup, you need to check for singularities at all points that lie above your singular point, not just the point with local coordinates (0,0). Fortunately, the equations you get from setting the partials to zero are not bad ... (or alternatively, some inspection should reveal the point $P$ where the equation will be in $\mathfrak{m}_P^2$).

Comment by Michael Joyce

2013-05-02T13:03:10Z

I read somewhere (on MO or MSE I think) that over time, a hyphenated word loses the hyphen over time as the word becomes more common. Originally, people used the word 'e-mail' but now 'email' is ubiquitous. The same thing is probably happening to quasifinite and quasiprojective; it is just a slower process because those words are not as widely used.

Comment by Michael Joyce

2013-04-17T02:43:13Z

I am generally considering an algebraically closed field, and I am fine with characteristic zero assumptions if necessary. I don't think reductive adds anything important, but I tried to cast as wide a net as possible while fishing for examples. I'll add an edit to say a little about motivation.

Comment by Michael Joyce

2013-04-14T15:01:02Z

The link in the OP's update does not work. (The correct link is below in Gil Kalai's answer.)

Comment by Michael Joyce

2013-04-05T12:53:19Z

The assertion formerly known as Theorem 2? With some LaTeX work, you could then use the Prince symbol to refer to it in the paper. en.wikipedia.org/wiki/…

Comment by Michael Joyce

2013-04-03T19:08:29Z

The idea of the projective plane is at least implicit in the work of the ancient Greek Pappus. It's not clear to me that one can say when projective space was developed. Anyway, it's still an interesting question, but I think it might be helpful to indicate exactly which aspect of projective space you are interested in. (I realize your second question does that, but I didn't know how to interpret your first question. Are you specifically interested in when people understood the generalization to $n$-dimensional space, or the first incarnations when $n = 2, 3$?)

Comment by Michael Joyce

2013-04-03T14:46:24Z

Start here? en.wikipedia.org/wiki/Projective_geometry

Comment by Michael Joyce

2013-03-15T17:48:20Z

I think the article of Haiman that you are thinking of is "Dual equivalence with applications, including a conjecture of Proctor", sciencedirect.com/science/article/pii/…

Comment by Michael Joyce

2013-02-15T12:37:09Z

Thanks! I've used \{ and \} successfully in the past, but it must have been in the comments.

Comment by Michael Joyce

2013-02-04T06:09:26Z

Please don't post a question on both this site and math stackexchange. Since this question is not research level, it is not appropriate for this site. For others: the question has been answered on math stackexchange.

Comment by Michael Joyce

2013-02-03T14:11:25Z

So I really should have ordered the list in the more fun manner of N-U-T. As far as I know, there is no use made of the ordering of this trichotomy in the study of spherical varieties, at least not explicitly.

Comment by Michael Joyce

2012-12-29T22:02:46Z

@darij grinberg: Only using simple transpositions defines the weak order on S_n, while using all transpositions defines the Bruhat order.

Comment by Michael Joyce

2012-12-13T12:39:19Z

You might start with Seshadri's "Standard Monomial Theory -- A Historical Account" (in volume 2 of his collected works) or Lakshmibai/Littelmann/Magyar's "Standard Monomial Theory and Applications" (available online).

Comment by Michael Joyce

2012-11-30T22:20:14Z

I don't see any reason that you should close this suggestion thread or those on other sites when MoMath opens. Presumably, even after the first exhibits go up you'll be changing the exhibits from time to time and occasionally want to design new ones.

Comment by Michael Joyce

2012-11-14T17:01:13Z

Also if you post the question on Math stackexchange, you should state what definition of Chern class you are using.

Comment by Michael Joyce

2012-11-14T17:00:22Z

This is a perfectly reasonable question, but not for this site. The best place for this question is math.stackexchange.com