## Chromotopy

### Topologized objects in algebraic topology

May 2nd, 2013

I attended MIT’s Talbot workshop last week, which was super fun and — at times — even productive. I had a couple facts about Morava E-theory come up repeatedly in side-conversations, and I think it’s worth sharing those with the rest of you now.

Here’s the point: continuous Morava E-theory is the only sort of Morava E-theory. Let me explain by taking a moment to recall where it comes from. The central thesis of chromatic homotopy theory is that the stable category is tightly bound to the moduli stack of 1-dimensional, commutative, smooth formal groups via the homology theory $MU_*$ of complex bordism. Generally, to any ring spectrum E we can associate a simplicial scheme given by the following fancy construction:

$\mathrm{Spec}\,\pi_* E \begin{array}{c}\leftarrow \\ \rightarrow \\ \leftarrow\end{array} \mathrm{Spec}\,\pi_* \left(\begin{array}{c}E \\ \wedge \\ E\end{array}\right) \begin{array}{c}\leftarrow \\ \rightarrow \\ \leftarrow \\ \rightarrow \\ \leftarrow \end{array} \mathrm{Spec}\,\pi_* \left(\begin{array}{c}E \\ \wedge \\ E \\ \wedge \\ E\end{array}\right) \cdots.$

The assignment $X \mapsto E_* X$ can be thought of as sending X to a quasicoherent sheaf over this simplicial scheme. In the case of MU, the associated simplicial scheme is equivalent to the moduli of formal groups, and various niceness results in algebraic topology say that this assignment is not too lossy — in many ways, the stable category behaves like $\mathrm{QCoh}(\mathcal{M}_{\mathbf{fg}})$.

Number theorists know quite a bit about formal groups, and we can shamelessly piggyback on their hard work to learn things about the stable category. For instance, after localizing at a prime, there are countably many geometric points on the moduli of formal groups, belonging to the Honda formal groups and enumerated by “height”. In stable homotopy theory, these geometric points are reflected by the following theorem:

(One of the Hopkins-Smith periodicity theorems:) There is a sequence of homology theories called Morava K-theories, which are complex oriented with formal group given by these Honda formal groups. Moreover, these give “exhaustive” list of field spectra, in the sense that every homology theory with natural Künneth isomorphisms is some kind of field extension of a Morava K-theory.

The next place to look for instruction from the number theorists after the points of $\mathcal{M}_{\mathbf{fg}}$ is in its local geometry. This analysis was carried out by Lubin and Tate, who found the following result:

(Lubin-Tate:) Every geometric point on $\mathcal{M}_{\mathbf{fg}}$ is smooth. The infinitesimal deformation space of the Honda formal group with height n has dimension (n-1).

This, too, is reflected in stable homotopy theory: associated to any formal group G over a perfect field there is a spectrum $E_n$, called Morava E-theory, whose coefficient ring is given by the infinitesimal deformation space of G.

Morava K-theories have lots of nice properties, largely owing to their Künneth isomorphisms, but Morava E-theories also have lots of nice properties from the perspective of structured ring spectra, largely owing to the fact that they are mixed- rather than positive-characteristic. Of course, the K- and E-theory of a particular formal group are closely related to one another, and so a lot of algebraic topology gets done by using whichever is appropriate and transferring results back and forth as necessary.

This all sounds great, but there’s a wrinkle: everything in the above story works perfectly when studying cohomology theories, but there are two sorts of Morava E-homology people consider: ordinary and continuous. These are defined respectively by the following two formulas:

$E_* X = \pi_* E \wedge X,$       $E^\vee_* X = \pi_* L_K (E \wedge X).$

The left-hand formula is exactly what you’d expect, so why are we studying the right-hand formula? After all, it’s not even properly a homology theory — localization is not expected to preserve infinite wedges!

The literal meaning “homology theory” aside, the right-hand formula is in fact the one we want. To start to see why, let’s recall the following formula for K-localization: if X is an E-local spectrum (and $X \wedge E$ is always so local), then we have the formula

$L_K X = \lim_I M^0(I) \wedge X,$

where $M^0(I)$ is a finite spectrum with the property $BP_* M^0(I) = BP_* / (p^{I_0}, v_1^{I_1}, \ldots, v_{n-1}^{I_{n-1}})$. It is, again, a consequence of the periodicity theorems that such spectra exist for $I \gg 0$, along with the expected quotient maps used to build the inverse system.

This formula indicates what’s really going on: the infinitesimal deformation space of a point is always defined as a formal scheme, i.e., as the formal colimit of a directed system of finite schemes, each of which captures a “nilpotent of degree m” piece of the deformation space, with m growing large. In the case of a smooth point, the deformation space is given by a power series ring, and this colimit amounts to remembering the adic topology on this ring. This is actually a huge deal — the formal spectrum of a power series ring (i.e., the spectrum with this topology taken into account) is what you’d expect: a single point, with structure sheaf at the point given by the power series ring. On the other hand, just the spectrum of a power series ring is terrifyingly large and repugnant — it has all sorts of prime ideals that you wouldn’t expect, on account of the existence of transcendental power series.

In turn, that’s what the extra K-localization is doing: it’s recalling that Lubin-Tate space (and hence Morava E-theory) is given as an ind-scheme (resp., as a pro-spectrum), and certainly we shouldn’t forget this crucial fact about our set-up. This extra step has all sorts of niceness consequences in algebraic topology; for instance, we have the following:

(Hovey-Strickland:) When X is a spectrum such that $E^* X$ is even-concentrated, then $K^* X$ is given by reduction at the maximal ideal: $K^* X = E^* X / I_n$. Conversely, when $K^* X$ is even-concentrated, then $E^* X$ is pro-free with this same reduction property.

There is an analogue of this property for continuous E-homology $E^\vee_*$ but not for $E_*$. In fact, the failure of this for the noncontinuous version is dramatic: its Bousfield class is given by a wedge of Morava K-theories, with heights ranging from 0 up to n. You can sort of see this coming — without continuity to control the image of the power series generators, you can imagine sending one of them to an invertible element in some target ring, thereby decreasing the height of the pulled-back formal group law. On the other hand, the Hovey-Strickland theorem says that the cohomological Bousfield classes of K- and E-theory coincide, as you would reasonably expect. That there is a Hovey-Strickland theorem for K-homology and continuous Morava E-homology says, among other things, that we’re specifically repairing this defect in the homological Bousfield class of E-theory.

OK, deep breath.

I think these are neat facts, coming from being careful about the insertion of formal geometry into algebraic topology. However, there’s a different fact that this perspective doesn’t quite jive with and which I would like to understand better: fracture squares. Chromatic fracture is a big deal in algebraic topology; its noble goal is to reconstruct higher chromatic spectra from chromatic layers and gluing data. It takes the form of the following pullback square:

$\begin{array}{ccc} L_{E(n)} X & \to & L_{K(n)} X \\ \downarrow & & \downarrow \\ L_{E(n-1)} X & \to & L_{E(n-1)} L_{K(n)} X,\end{array}$

implicitly noting that the Bousfield classes of E- and K-theory don’t depend upon the choice of formal group, but just of its height n. If you like, you can inductively expand this out entirely in terms of K-theoretic localizations, and you’ll find you’re taking a limit over a big punctured cube whose vertices are given by all the K-theoretic localizations applied in sequence, with variations on which localizations you skip. Moreover, this construction and theorem crucial — it’s how we chromatic homotopy theorists do everything, from understanding finite spectra to constructing TMF.

That’s why it’s a puzzle that this doesn’t seem to play nicely with formal geometry. Suppose that X is an E-theory; then one of the corners in this fracture cube takes the form $L_{K(t)} E_n$. The homotopy of this spectrum is computable, given noncanonically in choice of generators by:

$\pi_0 L_{K(t)} E_n = W(k)[\![u_1, \ldots, u_{n-1}]\!][u_t^{-1}]^\wedge_{(p, v_1, \ldots, u_{t-1})}.$

What the hell is this thing? The first thing to say is that this doesn’t have an interpretation in terms of classical formal geometry: we’re inverting something in the maximal ideal (which is not that frightening; consider building $\mathbb{Q}_p$ from $\mathbb{Z}_p$) and then recompleting against a nonclosed ideal (which is frightening; it means the completion map is not continuous in the respective adic topologies). You might also worry that I claimed that this presentation is noncanonical in choice of generators, then inverted a particular element — but a surprising fact is that the recompletion erases this choice of inversion. Selecting different generators will give isomorphic rings after passing to the recompletion. This indicates that we are, in some sense, performing a geometric operation.

But what operation and what geometry?

I’ve spoken a little bit to Nat Stapleton about this, who studies (among other things) “transchromatic character maps,” which use in an essential way this height-jumping feature of this strange localization of E-theory. We have some ideas (more accurately: he has some ideas) about what moduli problem this ring presents, but our phrasing of this is seriously crippled by the lack of an algebro-geometric framework in which to work. I’m at a loss as to what to do — we’ve spoken to a few algebraic geometers about this, and the most promising thing they’ve suggested has been Huber’s theory of adic spaces, but even this doesn’t quite accomplish the task at hand.

Who knows.

To wrap it up, topologized objects appear elsewhere in stable homotopy theory as well. For instance, work of Ando-Morava-Sadofsky and of Ando-Morava use pro-systems of Thom spectra to do some pretty cool transchromatic things. The Tate construction can also be thought of in such a way, as can parts of Lin’s theorem / the Segal conjecture. So, these sorts things are all around, and given their collective mysterious nature, I feel that we really don’t have a good way of thinking about them yet.

### Stanley symmetric functions

April 19th, 2013

Besides the original definition in terms of reduced words, the Stanley symmetric functions $F_w$ can be computed in at least three ways, which seem mysteriously different enough to me that it’s worth talking about all of them.

Combinatorics

It turns out that in the Schur expansion $F_w = \sum_{\lambda} a_{\lambda w} s_{\lambda}$, the coefficients $a_{\lambda w}$ are nonnegative integers, so it’s natural to ask for a combinatorial interpretation for them. Edelman and Greene gave a nice answer to this question. The column word of a semistandard tableau $T$ is the word gotten by reading up the first column, then the second column, and so on. Say $EG(w)$ is the set of semistandard tableaux whose column words are reduced words for $w$. Then $a_{\lambda w}$ is the number of tableaux in $EG(w)$ with shape $\lambda$.

For example, $Red(2143) = \{13, 31\}$, and each of these is the column word of exactly one tableau: $\begin{array}{cc} 1 & 3 \end{array}$ and $\begin{array}{c} 1\\ 3 \end{array}$ respectively, so $F_{2143} = s_{11} + s_{2}$. $Red(321) = \{121, 212\}$, but only the second of these can be the column word of a semistandard tableau, namely $\begin{array}{cc} 1 & 2\\ 2 & \end{array}$, so $F_{321} = s_{21}$.

In fact Edelman-Greene do better than this. Previously I mentioned that the equality $|Red(w)| = \sum_{\lambda} a_{w\lambda} f^{\lambda}$ falls out of the definition of $F_w$, where $f^{\lambda}$ is the number of standard tableaux of shape $\lambda$. Given the interpretation of $a_{w\lambda}$, this  suggests there should be a bijection

$Red(w) \leftrightarrow \{(P, Q) : P \in EG(w), \text{shape}(P) = \text{shape}(Q), Q \text{ standard} \}$.

Edelman-Greene provide just such a bijection. If you know the Robinson-Schensted(-Knuth) correspondence, this should look familiar—Edelman-Greene’s bijection is very similar, and includes Robinson-Schensted as the special case $w = 2143\cdots (2n)(2n-1)$.

This also solves the problem of sampling uniformly from $Red(w)$, as long as we can compute $EG(w)$. For example, take the special case $w = n(n-1)\cdots 1$, where $EG(w)$ has just one element $P_0$ (because $F_w = s_{(n-1, n-2, \ldots, 1)}$). It turns out to be easy to uniformly choose a standard tableau $Q$ of a particular shape (using the “hook walk” algorithm), so choose $Q$ with the same shape as $P_0$ and apply the Edelman-Greene bijection to the pair $(P_0, Q)$ to obtain a reduced word.

Representation theory

The Edelman-Greene bijection shows that the coefficients $a_{w\lambda}$ are nonnegative integers. Since the Schur functions $s_{\lambda}$ are irreducible characters for $GL(E)$ (if $\text{dim}(E)$ is large enough), this means $F_w$ is the character of a $GL(E)$-module. This module turns out to have a nice description not relying on knowing its decomposition into irreducibles.

Here’s how to get the module $E^{\lambda}$ whose character is $s_{\lambda}$ (called a Schur module). Start with the left $GL(E)$-module $E^{\otimes n}$, where $n = |\lambda|$. This is also a right $S_n$-module, permutations acting by permuting tensor factors. Construct the Young symmetrizer $c_{\lambda}$, a member of the group algebra $\mathbb{C}[S_n]$. Then the Schur module $E^{\lambda}$ is $E^{\otimes n}c_{\lambda}$.

The construction of the Young symmetrizer $c_{\lambda}$ uses the Ferrers diagram of $\lambda$, but only actually relies on having a set of boxes to put numbers in, not that the boxes are arranged in any particular way. That is, any finite set $D \subset \mathbb{N}^2$ has a Young symmetrizer $c_D$ using the same definition, and so there’s a $GL(E)$-module $E^D = E^{\otimes n}c_D$.

The Stanley symmetric function $F_w$ turns out to be the character of one of these guys. The inversion set of a permutation $w$ is $I(w) = \{(i,j) : i < j, w(i) > w(j)\}$. Then $F_w$ is the character of $E^{I(w^{-1})}$. [I'm not sure who to credit this to... Kraśkiewicz and Pragacz? Let's say yes, Poles unite!!]

For example,

$I(321) = \{(1,2),(1,3),(2,3)\} = \begin{array}{ccc} \cdot & \square & \square \\ & \cdot & \square \\ & & \cdot \end{array}$

is more or less the Ferrers diagram of $(2,1)$, so we recover again that $F_{321} = s_{21}$. The equality $F_{2143} = s_{11} + s_{2}$ from this point of view ends up being the classic decomposition of 2-tensors into symmetric and antisymmetric parts, $E \otimes E \simeq Sym^2(E) \oplus \bigwedge^2(E)$.

Geometry

Schubert calculus, in modern terms, amounts to doing calculations in the cohomology ring of Grassmannians or flag varieties (or more generally, a semisimple algebraic group mod a parabolic subgroup). Take $B^+, B^-$ the subgroups of upper and lower triangular matrices, respectively, in $GL_n$. A (complete) flag in $\mathbb{C}^n$ is a sequence of subspaces $0 = V_0 \subseteq V_1 \subseteq \cdots \subseteq V_n = \mathbb{C}^{n}$ with $\text{dim}(V_i) = i$. The collection of flags in $\mathbb{C}^n$ is naturally in bijection with the flag variety $Fl(n) = GL_n / B^+$, a projective variety.

The LU decomposition (or LPU) gives the Bruhat decomposition of $GL_n$: the disjoint union $GL_n = \bigcup_{w \in S_n} B^- w B^+$, interpreting $w$ as a permutation matrix. This descends to a decomposition $Fl(n) = \bigcup_{w \in S_n} B^- w B^+ / B^+$. Each $B^- w B^+ / B^+$ is a Schubert cell $X_w^o$, homeomorphic to some $\mathbb{C}^k$; the Schubert variety $X_w$ is the closure of $X_w^o$. The Schubert cells form a CW decomposition of $Fl(n)$, and have even real dimensions, so the classes $[X_w]$ Poincaré dual to the Schubert varieties form a $\mathbb{Z}$-basis of $H^*(Fl(n), \mathbb{Z})$.

As for the ring structure, by realizing $Fl(n)$ as the total space of the top of a tower of projective bundles over $\mathbb{P}^1$, one can compute that $H^*(Fl(n))$ is a certain quotient of $\mathbb{Z}[x_1, \ldots, x_n]$. Lots of classical enumerative geometry questions can be phrased in terms of counting intersection points between various Schubert varieties, so one should try to find polynomials representing the Schubert classes $[X_w]$ in this picture of $H^*(Fl(n))$.

Lascoux and Schützenberger found a nice set of such polynomials, the Schubert polynomials $\mathfrak{S}_w$. They have the nice property of being stable under the inclusions $Fl(n) \to Fl(n+1)$, meaning that for $w \in S_n$, $\mathfrak{S}_w$ represents $[X_w]$ in $H^*(Fl(N))$ whenever $N \geq n$, if we view $w$ as a permutation of $[N]$ fixing $n+1, \ldots, N$.

Schubert polynomials enjoy another kind of stability. Write $1^m \times w$ for the permutation $1\cdots m (w(1)+m)(w(2)+m) \cdots$. It turns out that as $m \to \infty$, $\mathfrak{S}_{1^m \times w}$ converges to a power series in $x_1, x_2, \ldots$. This power series is exactly $F_{w^{-1}}$!

### Determinantal K-theory

April 4th, 2013

Craig Westerland and I have been going around giving talks about something we stumbled across in K(n)-local homotopy theory and can’t yet really explain. It’s a really neat construction, and I think it’s mind-blowing that it works as well as it does, and I can’t hope to tell everyone about it in a talk — so instead I’ll blog about it a little. There are some serious words coming up, so hold on to your hats.

This story picks up where the previous post about spectral tangent spaces left off: there’s some machine which swallows spectra-with-diagonals whose cohomology looks like that of $\mathbb{C}\mathrm{P}^\infty$ and produces elements of the K(n)-local Picard group. In fact, I can say how this machine works: suppose you’re given a K-algebra A and a scheme-theoretic point $x: A \to K$. The kernel of this map is the ideal I, which is thought of as functions vanishing at x. This ideal receives a multiplication map $I \otimes_K I \to I$ restricting the one for A, and the quotient (or cokernel) is the definition of the cotangent space: $T^*_x A = I / I \otimes_K I$. We can say all these words for spectra too, but the theory works out better if we use coalgebras rather than algebras, essentially because the Spanier-Whitehead duals of infinite objects are complicated. So: we have a pointed coalgebra spectrum $\eta: \mathbb{S} \to C$. This has a cofiber M, which is a C-comodule spectrum. It also supports a projection of the diagonal $M \to M \square_C M$, the fiber of which we define to be the tangent space $T_\eta C$ at $\eta$. The effort in this construction comes from deciding what the cotensor product $M \square_C M$ should mean, and how to control it once the decision is made — but it turns out that this is accomplishable.

So, like I said in the previous post, there are two obvious choices of spectra you can feed into this: $\Sigma^\infty_+ \mathbb{C}\mathrm{P}^\infty$ and $\Sigma^\infty_+ K(\mathbb{Z}, n+1)$. In the first case, the spectrum you get out is the 2-sphere, and in the latter case, the spectrum you get out is a nonstandard invertible spectrum called the “determinantal sphere,” which is meant to describe its image as a line bundle over Lubin-Tate space:

$\mathcal{E}_n(T_+ \Sigma^\infty_+ K(\mathbb{Z}, n+1)) \cong \Omega^{n-1}_{LT_n / \mathbb{Z}_p}.$

It’s not super important what this actually means — mostly it means that for $n > 1$, it simply cannot be a standard sphere.

So, that catches us up with the previous post. Here’s the first remarkable fact: this process is iterable. The definition of cotangent space studies 1-jets of functions, but we can consider pieces of m-jets instead, i.e., the quotients $I^{\otimes_K m} / I^{\otimes_K m+1}$, or even the spectral fibers $A_m = \mathrm{fib}\;M^{\square_C m} \to M^{\square_C m+1}$. It turns out that there is a K(n)-local equivalence $A_m = (A_1)^{\wedge m}$ describing these further quotients as smash powers of the first one. What’s interesting about this is most visible in the case $C = \mathbb{C}\mathrm{P}^\infty$: these spectra $M^{\square_C m}$ form a sort of cofiltration of C whose filtration quotients are given by $A_m = (A_1)^{\wedge m} = \mathbb{S}^{2m}$. What that must mean is that we’re picking up the cellular filtration of $\mathbb{C}\mathrm{P}^\infty$. Of course, this machine works just as well with $C = K(\mathbb{Z}, n+1)$, and once again I encourage you to think of this jet decomposition as a kind of cellular filtration of C — but this time “cell” is relaxed to mean a disk attached along an arbitrary element of the K(n)-local Picard group, rather than merely a standard sphere. And this is really interesting — the classical cellular structure of $K(\mathbb{Z}, n+1)$ is dismally complicated, and that of its K(n)-localization is even worse, but if you enlarge your viewpoint a little bit things turn out to simplify dramatically.

Having made this comparison between these two spectra, we may as well keep going. The inclusion of the 2-skeleton into $\mathbb{C}\mathrm{P}^\infty$ has another name: it is the Bott class in homotopy $\beta: \mathbb{C}\mathrm{P}^1 \to \mathbb{C}\mathrm{P}^\infty$. Using the jet decomposition of $K(\mathbb{Z}, n+1)$, we get a “Bott class” there too:

$\beta: T_+ \Sigma^\infty_+ K(\mathbb{Z}, n+1) \to \Sigma^\infty_+ K(\mathbb{Z}, n+1),$

which is just the inclusion of the fiber $T_\eta C \to M$ from before. What good is identifying the Bott element? — well, Snaith’s theorem says that you can use it to build complex K-theory:

(Snaith:) As ring spectra, $\Sigma^\infty_+ \mathbb{C}\mathrm{P}^\infty[\beta^{-1}] \simeq KU.$

Of course, we can do this too: we define a spectrum $R_n = \Sigma^\infty_+ K(\mathbb{Z}, n+1)[\beta^{-1}]$, which I strongly recommend thinking of as a K-theory spectrum — so strongly that I’m going to call it “determinantal K-theory”. I’ll try to convince you that this is a good name in a moment. For now, let’s justify the name by noting that when $n = 1$ it is precisely complex K-theory.

Craig Westerland has a remarkable theorem placing this object in classical homotopy theory:

(Westerland:) There is an equivalence $R_n \simeq E_n^{hS\mathbb{S}_n}$ of ring spectra, where $S\mathbb{S}_n$ is the subgroup of “special” elements of the Morava stabilizer group. These are the elements in the kernel of the determinant map $\mathbb{S}_n \to \mathbb{Z}_p^\times$ which arises from considering $\mathbb{S}_n$ as acting on a certain division algebra as its units.

This is neat enough even if you don’t know about classical Picard computations — for instance, this says that $R_n$ receives the structure of an $E_\infty$ ring spectrum — but the story turns out to be better than that. Since $S\mathbb{S}_n$ is such a large subgroup, there are very few automorphisms of $R_n$ left. Specifically, taking $\gamma$ to be a topological generator of $\mathbb{Z}_p^\times$, there is a fiber sequence

$(L_{K(n)} \mathbb{S}^0 =) E_n^{h\mathbb{S}_n} \to E_n^{hS\mathbb{S}_n} \xrightarrow{\psi^\gamma - 1} E_n^{hS\mathbb{S}_n},$

identifying $R_n$ as “half” of the K(n)-local sphere in this precise sense.

… And then you refuse to stop there. The space BU is constructed from the spectrum KU by the formula $BU = \Omega^\infty KU \langle 1 \rangle$, and so we too can build an analogue $W_n = \Omega^\infty R_n \langle 1 \rangle$. This turns out to have a natural map $J_n: W_n \to BGL_1\mathbb{S}^0$, generalizing the classical J-homomorphism when n is set to 1. It even detects a family of elements identical in shape to the image of J, but in the homotopy groups graded along powers of the determinantal sphere. It also yields a Thom spectrum $X_n = \mathrm{Thom}(J_n)$, in analogy to MU, and there is then a theory of orientations. The spectrum $R_n$ also has the same universality property as complex K-theory: it is initial among ring spectra which are multiplicatively oriented, where “orientation” is taken relative to this new replacement for MU. And the list goes on.

To me, this is really impressive and exciting — but it’s also young, which means there are loads of things unanswered and pending. I’ll point out two big ones, but there are plenty of others too if you go looking.

1. There is no mention of geometry in this discussion. In particular, there is no known analogue of the spaces $BU(m)$ for $m \neq 1, \infty$ — Snaith’s theorem lets us make the jump from knowing about “line bundles” to knowing about virtual vector bundles with no intermediate geometric step. This is cool, but it’s also sort of bad news — it cuts us off from the enormous sector of mathematics that has developed around classical K-theory, and it also makes us rather blind about what to do next. Curiously, it’s rather easy to construct analogues of the spaces $\Omega SU(m)$ — but doing so gives no indication of any underlying geometry.
2. There’s a second filtration that we may have accidentally confused with the skeletal filtration. Namely, there are equivalences $\mathbb{C}\mathrm{P}^\infty \simeq BU(1)$ and $\mathbb{R}\mathrm{P}^\infty \simeq BO(1)$, and the first filtration quotients of the bar construction for these two spaces are $S^2 \simeq \Sigma U(1)$ and $S^1 \simeq \Sigma O(1)$ respectively. So, we might further guess that $K(\mathbb{Z}, n+1)$ can be written as $BG(1)$ in the K(n)-local stable category in a nonstandard way — it may be possible to pick $G(1) = \Sigma^{-1} T_+ \Sigma^\infty_+ K(\mathbb{Z}, n+1)$ and produce an appropriate $A_\infty$ multiplication on it, i.e., there may be interesting “elements of Hopf invariant one” that live on nonstandard spheres. That would certainly be a push toward becoming geometrically informed about $R_n$.

In any case, there seems to be an elephant out there, and we seem to have gotten a pretty good hold of its tail. It will be really cool to see how this all sorts out in the distant future.

### The (very nice) Bousfield Lattice of Harmonic Spectra

April 1st, 2013

So this is my first post here on Chromotopy (thanks to Matt Pancia for making me aware of this blog, and Eric Peterson for letting exercise my typing fingers!). Just thought I might mention a little doodle about Bousfield lattices that occurred to me recently. It’s nothing too complicated, and it’s kind of cute, so I’ll include the proof. This is something that I realized while I was futzing around with general Bousfield lattices, notions of Stone duality (i.e. categorical anti-equivalences between certain species of lattice and certain species of topological space). Ultimately, I suspect it follows from one or two of the many beautiful theorems hidden away in papers like “Axiomatic Stable Homotopy Theory” and other stuff by Hovey, Strickland and Palmieri.

Here’s the point: in Mark Hovey and Neil Strickland’s beautiful (and incredibly dense) paper “Morava K-theories and Localisation” they show that the Bousfield lattice of the $E(n)$-local category of spectra is precisely the finite Boolean algebra generated by $n$ objects, specifically the Bousfield classes of Morava K-theories, $\langle K(n)\rangle$.  The natural next question, for me at least, is what the Bousfield lattice of the harmonic category (i.e. the $p$-local stable homotopy category localized at the infinite wedge of Morava K-theories $\bigvee_n K(n)$) looks like (let’s denote this category by $\mathcal{H}$).  And it turns out, it’s precisely what one would expect! That is, it’s the colimit of the Bousfield classes of the $E(n)$-local categories, i.e. the infinite Boolean algebra generated by a countable number of minimal elements.

How do we know this? Well, what we really want to do is determine the harmonic Bousfield class of a harmonic spectrum $X$, which I’ll denote by $\langle X\rangle_H$.  What does that really mean? It means that $\langle X\rangle_H=\{ Y\in\mathcal{H}:Y\wedge X\simeq \ast\}$.  If you’re quick (which I am not), you certainly already see how to do this. If you’re like me, it might take you a few days, so since it’s not that big of a deal, you should just keep reading.

Let $supp(X)=\{K(n):X\wedge K(n) \not\simeq \ast\}$ and $cosupp(X)=\{K(n):X\wedge K(n)\simeq \ast\}$ .  What we’re going to do is show that $\langle X\rangle_H=\bigvee_{K(n)\in supp(X)} \langle K(n)\rangle$.  Suppose that $Y\in\langle X\rangle_H$.  So, $Y\wedge X\simeq\ast$, and imortantly, $Y\wedge X\wedge K(n)\simeq\ast$.  In other words, $Y\in\langle X\wedge K(n)\rangle_H=\langle K(n)\rangle_H$. Where this last equality follows from the fact that $K(n)\wedge X=\vee\Sigma^{d_i} K(n)$ for some collection of $d_i$. Hence, $Y\in\langle \bigvee_{supp(X)}K(n)\rangle=\bigvee_{supp(X)}\langle K(n)\rangle$.

Now, suppose that $Y\in\langle \bigvee_{supp(X)}K(n)\rangle$.  Then we want to show that $Y\wedge X\simeq\ast$.  So, for every $K(n)\in supp(X)$, it’s clear that $Y\wedge X\wedge K(n)\simeq\ast$.  But for every $K(m)\in cosupp(X)$, we have that $Y\wedge X\wedge K(m)\simeq\ast$ also! So, since $X\wedge Y$ is harmonic, but $X\wedge Y\wedge K(n)\simeq\ast$ for every $n$, it’s also contractible!

One remark: I’ve neatly brushed under the rug here that colocalizing subcategories (e.g. harmonic spectra) may not be closed under smash product (I don’t think).  So, I really should have written $\wedge_H$ or something every time I wrote $\wedge$, but it works out fine if you just assume that after I took the smash product, I localized at the harmonic category again.

Anyway, the reason I think this is interesting is that this lattice has a really nice Zariski spectrum, and I first started looking at it hoping that it might approximate the Bousfield lattice of $p$-local spectra in some nice way.  But as far as I can tell, it’s just too simple (you never get something for nothing, I suppose).  I was hoping to make something of some ideas of Jack Morava regarding sheaves of spectra over the Bousfield lattice, but it may be that Bousfield lattices are just too coarse to do this kind of thing, or rather, tensor-triangulated categories are too coarse. We really need to rewrite the HPS paper I referenced above in the language of infinity categories and derived algebraic geometry.

It’s also interesting to note that the little proof above implies that the telescope conjecture holds inside the category of harmonic spectra.  Obviously this doesn’t really say anything about the telescope conjecture outside this category. It’s also known, for instance, that the telescope conjecture holds inside the category of $BP$-local spectra. However, the Bousfield lattice in that category is not known. It’s structure is the content of a conjecture from Doug Ravenel’s well known ’84 paper that is still open!

I hope to write more later about Stone duality for lattices/locales and Bousfield lattices in general.

### Reduced words

February 18th, 2013

This past summer I helped out a bit on an REU project, on extending a few results and conjectures from this paper. Those results/conjectures are both amazing and easy to describe (if not to prove), so I want to do that here. It’s also a nice excuse to talk about the story of reduced words for permutations.

Let $s_i$ be the transposition $(i\, \,\,\,i+1)$. A reduced word for a permutation $w$ is a sequence $a = a_1 \cdots a_{\ell}$ of integers such that $w = s_{a_1} \cdots s_{a_{\ell}}$ and $\ell$ is minimal. That is, a reduced word is a minimal way of getting from $12\cdots n$ to $w$ by repeatedly interchanging adjacent entries.

Examples

There are 2 reduced words for $w = 2143$:

$1234 \,\xrightarrow{1}\, 2134 \,\xrightarrow{3}\, 2143 \,\leadsto\, a = 13$

$1234 \,\xrightarrow{3}\, 1243 \,\xrightarrow{1}\, 2143 \,\leadsto\, a = 31$

$w = 321$ has 2 also: $a = 121$ and $212$.

A little thought will show that the length of a reduced word for $w$ is the same as the length of $w$, the number of inversions in $w$: pairs $i < j$ such that $w_i > w_j$ . It’s also not hard to see that reduced words always exist, i.e. that the symmetric group is generated by adjacent transpositions.

The element of $S_n$ of greatest length is the reverse permutation $w_0 = n(n-1)\cdots 21$ , having length ${n \choose 2}$ . The basic question of Angel et al. is: what does a random reduced word for $w_0$ look like? We’ll look at two ways of visualizing a reduced word. First, think of a reduced word $a = a_1 \cdots a_{\ell}$ as a sequence of permutations:

$1,\quad s_{a_1},\quad s_{a_1} s_{a_2},\quad \ldots,\quad s_{a_1} \cdots s_{a_\ell}$ .

For each entry $1, \ldots, n$, track its position in each permutation in the sequence. Since at each step we just switch two adjacent entries, positions change by at most 1 each time, so we can hope to get nice-looking piecewise curves. To be precise, for each $i$, draw the path which connects the points

$(0, i), (1, s_{1}(i)), (2, s_{a_1} s_{a_2}(i)), \ldots$ .

Here’s what you get, for $n = 500$ (having drawn only some of the paths, to reduce clutter):

Are these… sine waves?

Let’s press on. Consider the same sequence of permutations

$1,\quad s_{a_1},\quad s_{a_1} s_{a_2},\quad \ldots,\quad s_{a_1} \cdots s_{a_\ell}$ .

but now draw their graphs, as functions $\{1, \ldots, n\} \to \{1, \ldots, n\}$ , in sequence. In the following picture we do that and then go back to the identity.

!!!

Angel et al. don’t quite prove that you get these sine wave and rotating disk pictures in the limit, but they do give an attractive geometric interpretation of reduced words which should explain the pictures, in terms of the permutahedron. Instead of that, though, I’ll talk about how one samples reduced words uniformly at random to create these pictures. This might not sound so exciting, but in fact leads to some rather nice mathematics.

The obvious thing to do to choose a reduced word $a$ for $w$ is start with $12\cdots n$ , choose a random adjacent pair which isn’t an inversion, swap it, and repeat until you’re at $w$. Unfortunately, this doesn’t give a uniform distribution (try it for $w = 4321$). To understand how Angel et al. solve this problem, we’ll go back to the 80s when Stanley was thinking about counting reduced words. First, however, we need some basics about symmetric functions.

Symmetric functions

A symmetric function is a formal power series $f(x_1, x_2, \ldots)$ in infinitely many variables, with integer coefficients, which remains unchanged under permutation of the variables: if $\sigma \in S_n$ , then $f(x_{\sigma(1)}, x_{\sigma(2)}, \ldots) = f(x_1, x_2, \ldots)$ . [We should also assume $f$ has terms of only finitely many different degrees, so that the ring of symmetric functions is graded by degree]. For example:

$x_1 + x_2 + x_3 + \cdots$  or  $x_1 x_2 + x_1 x_3 + x_2 x_3 + \cdots\,$ ;

more generally, the elementary symmetric function $e_k = \sum_{1 \leq i_1 < \cdots < i_k} x_{i_1} \cdots x_{i_k}$ , the sum of all squarefree degree $k$ monomials.

Here's one reason to care about symmetric functions: they're characters of $GL_n$-modules. Suppose $E$ is a finite-dimensional complex $GL_n(\mathbb{C})$-module, and let $T$ be the subgroup of diagonal matrices in $GL_n$ . To avoid awful things let's assume $E$ is really a representation of $GL_n$ as an algebraic group, meaning that the defining map $\rho : GL_n \to GL(E)$ is a map of varieties. Given $X \in T$ with diagonal entries $x_1, \ldots, x_n$ , the character of $E$ is $\chi_E(x_1, \ldots, x_n) = \text{tr} \rho(X)$ . Basic representation theory (Schur's lemma) shows this is a symmetric (Laurent1) polynomial in $x_1, \ldots, x_n$ .

For example, take $E = \bigwedge^k \mathbb{C}^n$ , and a basis $e_1, \dots, e_n$ of $\mathbb{C}^n$ . Any basis vector $e_{i_1} \wedge \cdots \wedge e_{i_k}$ for $E$ is an eigenvector for $X$ , with eigenvalue $x_{i_1} \cdots x_{i_k}$ . Now $\chi_E$ is the sum of all these eigenvalues, which is just the elementary symmetric function $e_k(x_1, \ldots, x_n)$ . It's similarly easy to see that the character of $Sym^k \mathbb{C}^n$ is the homogeneous symmetric function

$h_k = \sum_{1 \leq i_1 \leq \cdots \leq i_k \leq n} x_{i_1} \cdots x_{i_k}$ .

[The fact that we now just have finitely many variables can be safely glossed over.]

From this point of view, there are some obvious symmetric functions to look at: the characters of simple $GL_n$-modules. These turn out to be naturally indexed by partitions---finite integer sequences $\lambda = (\lambda_1 \geq \cdots \geq \lambda_{\ell})$, where $\ell \leq n$ . Say $\lambda$ is a partition of $n$ if $|\lambda| = \sum_i \lambda_i = n$. The Ferrers diagram of a partition $\lambda$ is gotten by drawing a left-justified row of $\lambda_i$ boxes for each $i$ , starting at the top and going down:

semistandard tableau (plural tableaux) of shape $\lambda$ is a filling of the boxes of the Ferrers diagram of $\lambda$ with positive integers which is strictly increasing down columns, weakly increasing rightward across rows:

To a semistandard tableau $T$, associate a monomial $x^T = \prod_{i \in T} x_i$, the product of $x_i$ as $i$ ranges over the entries of the tableau. For the tableau above, this monomial is $x_1^2 x_2^2 x_3 x_4^3 x_6$. Now given a partition $\lambda$, the Schur function $s_{\lambda}$ is

$s_{\lambda} = \sum_T x^T$,

where $T$ ranges over all semistandard tableaux of shape $\lambda$. It's less than obvious from this definition that $s_{\lambda}$ is symmetric, but it is. The Schur functions $s_{\lambda}(x_1, \ldots, x_n)$ as $\lambda$ runs over partitions with length at most $n$ turn out to be (almost1) exactly the irreducible characters of $GL_n$.

Examples

If $\lambda = (k)$, a semistandard tableau of shape $\lambda$ is a weakly increasing sequence $1 \leq i_1 \leq \cdots \leq i_k$. The Schur function $s_{(k)}$ is $h_{(k)}$ above, associated to $\text{Sym}^k \mathbb{C}^n$.

If $\lambda = (1, \ldots, 1) = (1^k)$, a semistandard tableau of shape $\lambda$ is a strictly increasing sequence $1 \leq i_1 < \cdots < i_k$, and $s_{(1^k)} = e_k$, associated to $\bigwedge^k \mathbb{C}^n$.

If $\lambda = (2,1)$, there are 2 semistandard tableaux with entries at most 2,

$\begin{array}{cc} 1 & 1\\ 2 & \, \end{array}\quad \begin{array}{cc} 1 & 2\\ 2 & \, \end{array}$,

and so $s_{(2,1)}(x_1, x_2) = x_1^2 x_2 + x_1 x_2^2$. To locate the corresponding $GL_2(\mathbb{C})$ module, do a bit of algebra:

$s_{(2,1)}(x_1, x_2) = (x_1 + x_2)(x_1 x_2) = h_1 e_2$,

so the module should be $\mathbb{C}^2 \otimes \bigwedge^2 \mathbb{C}^2$.

One last fact that we'll need, which follows from the realization of Schur functions as irreducible characters: as $\lambda$ ranges over all partitions of $n$, the Schur functions $s_{\lambda}$ form a $\mathbb{Z}$-basis of the degree $n$ symmetric functions.

Stanley symmetric functions

Let's get back to reduced words. If $\lambda$ is a partition of $n$, meaning that $|\lambda| = \sum_i \lambda_i = n$, a standard tableau of shape $\lambda$ is a semistandard tableau of shape $\lambda$ with entries $1, \ldots, n$. For example, $(2,1)$ has two standard tableaux,

$\begin{array}{cc} 1 & 2\\ 3 \end{array}\quad \begin{array}{cc} 1 & 3\\ 2 \end{array}$.

Stanley noticed the following coincidence: for $1 \leq n \leq 6$, the number of reduced words for $n(n-1)\cdots 21$ matches the number of standard tableaux of the staircase shape $\delta_n = (n-1, n-2, \ldots, 1)$; we've seen this for $n = 3$. The numbers are 1, 1, 2, 16, 768, 292864, so this is pretty convincing.

Now we define a somewhat mysterious-looking symmetric function. The Stanley symmetric function of a permutation $w$ is

$F_w = \sum_{a \in \text{Red}(w)} \sum_{i \in C(a)} x_{i_1} \cdots x_{i_\ell}$,

where $\text{Red}(w)$ is the set of reduced words of $w$, and $C(a)$ is the set of integer sequences $1 \leq i_1 \leq \cdots \leq i_\ell$ such that if $a_j > {j+1}$, then $i_j < i_{j+1}$. Here $\ell$ is the length of $w$.

[Where does this come from? It seems plausible that Stanley was aiming for $F_{w_0} = s_{\delta_n}$, for reasons which will become clear below, and already knew about similar "quasisymmetric" expansions for Schur functions.]

Example

Take $w = 2143$, with reduced words $31, 13$. Then $C(13) = \{1 \leq i \leq j\}$ and $C(31) = \{1 \leq i < j\}$, so

$F_{2143} = h_2 + e_2 = s_{(2)} + s_{(1,1)}$.

In this example $F_w$ is visibly symmetric, which is true in general, if not at all obvious.

Here's the key enumerative fact. Right from the definitions we see that the coefficient of $x_1 \cdots x_n$ in $s_{\lambda}$ is the number of standard tableaux of shape $\lambda$; call it $f^{\lambda}$. On the other hand, $12\cdots \ell \in C(a)$ for any reduced word $a$, and so the coefficient of $x_1 \cdots x_\ell$ in $F_w$ is the number of reduced words of $w$. So, if we write

$F_w = \sum_{\lambda} a_{\lambda} s_{\lambda}$,

then $|\text{Red}(w)| = \sum_{\lambda} a_{\lambda} f^{\lambda}\,$!

Of course this isn't so useful without a reasonable way of getting at the Schur decomposition of $F_w$. This post is long enough, so next time I'll describe three such ways: one combinatorial (which will also solve the problem of sampling from $\text{Red}(w_0)$ uniformly), one representation-theoretic (since $F_w$ is a symmetric function, maybe it's the character of some module???), and one based on the geometry of Schubert varieties.

Notes

1 Here "Laurent" and "almost" have the same explanation: the 1-dimensional representations of $GL_n$ where $A$ acts as multiplication by $\det(A)^{-k}$ for some positive integer $k$.