Recent MathOverflow Questions

How should I think about the Grothendieck-Springer alteration?

Math Overflow Recent Questions - Sat, 11/03/2018 - 12:07

Given a simple complex Lie algebra $\mathfrak{g}$, recall the Springer resolution of its nilpotent cone $\widetilde{\mathcal{N}}\to \mathcal{N}$. Several times I have seen someone explaining Springer theory in terms of perverse sheaves, and each time the existence of the Grothendieck-Springer alteration $\pi: \widetilde{\mathfrak{g}}\to\mathfrak{g}$ along with the diagram $$\require{AMScd}\begin{CD} \widetilde{\mathcal{N}} @>>> \widetilde{\mathfrak{g}} @>>>\mathfrak{t}\\ @VVV @VVV @VVV\\ \mathcal{N} @>>> \mathfrak{g}@>>>\mathfrak{t}/W \end{CD}$$ is magically pulled out of a hat (eg: There also exist this other thing that...), where $\mathfrak{t}$ is the universal Cartan and $W$ is the Weyl group. Then one uses the fact that $\pi$ is a small map, giving an IC sheaf $\pi_\ast\underline{\mathbb{Q}}_\widetilde{\mathfrak{g}}$ with a natural $W$-action, which in turn induces a $W$-action on the Springer sheaf by some functoriality. I find this unsatisfying because it seems like $\widetilde{\mathfrak{g}}$ is kept mysterious.

My vague question is how should I think about the Grothendieck-Springer resolution and what is its role in modern representation theory? I know this is not a good question, so let me try to refine it by asking the two following questions.

1) Is there a broader theoretical context to fit the above diagram into where I am given a resolution of singularities $X_0\to Y_0$ (maybe with $X$ symplectic?), and can find a smooth family $X\to T$ and a proper map of smooth varieties $X\to Y$ fitting into the diagram $$\require{AMScd}\begin{CD} X_0 @>>> X \\ @VVV @VVV\\ Y_0 @>>> Y, \end{CD}$$ or is the Springer map special in a sense I don't understand?

Aside from applications to proving a generalized Springer correspondence, are there other examples where the existence and properties of this remarkable space are used in representation theory?

2) What are other applications of the Grothendieck-Springer resolution?

For example, the Springer resolution can be interpreted as a moment map, and David Ben-Zvi's answer to this question shows how this may be interpreted as the semiclassical shadow to Beilinson-Bernstein localization. Is there an analogous quantization of $\widetilde{\mathfrak{g}}\to \mathfrak{g}$? EDIT: I would be particularly interested in applications which are not so closely connected with the Springer resolution.

I'll stop here, since I have probably already asked too many questions. I would greatly appreciate any references to a modern understanding of $\widetilde{\mathfrak{g}}$.

Is $\in$-induction provable in first order Zermelo set theory?

Math Overflow Recent Questions - Sat, 11/03/2018 - 11:54

Are there models of first order Zermelo set theory (axiomatized by: Extensionaity, Foundation, empty set, pairing, set union, power, Separation, infinity) in which $\in$-induction fail?

I asked this question on Mathematics Stack Exchange here 4 days ago, to receive no answer yet, I thought this question already has a well known answer that is somehow evading me. Seeing that nobody answered thus far, makes me wonder if this is really an elementary issue? The whole issue is whether the proof that Foundation implies $\in$-induction necessitate existence of transitive closures for all sets, and since first order Zermelo doesn't prove the existence of transitive closures for all sets, then this raised this issue in my mind.

How to compute the inverse of a quantum determinant?

Math Overflow Recent Questions - Sat, 11/03/2018 - 10:26

Let $X=(x_{ij})_{mn}$ be a quantum matrix with the commutation relations between entries: \begin{alignat*}{2} & x_{ij} x_{il} = q x_{il} x_{ij}, && j < l, \\ & x_{ij} x_{kj} = q x_{kj} x_{ij}, && i<k, \\ & x_{ij} x_{kl} = x_{kl} x_{ij}, && i<k, j>l, \\ & x_{ij} x_{kl} = x_{kl} x_{ij} + (q - q^{-1})x_{il}x_{kj}, \quad && i < k, j < l. \end{alignat*} For $n \in \mathbb{Z}_{>0}$, denote $[n]=\{1, \ldots, n\}$. For any $I \subset [m]$, $J \subset [n]$, $|I|=|J|=l \in \mathbb{Z}_{>0}$, a quantum minor $\Delta_{I,J}$ is defined as follows \begin{align*} \Delta_{I,J} = \sum_{\sigma \in S_l} (-q)^{\ell(\sigma)} x_{i_1, j_{\sigma(1)}} \cdots x_{i_l, j_{\sigma(l)}}, \end{align*} where $\{i_1 < \cdots < i_l\} = I$, $\{j_1< \cdots < j_l\}=J$, and $\ell(\sigma)$ is the length of the permutation $\sigma$.

When $m=n$ and $I=[n]$, $\Delta_{I,I}=\det_q(X)$ is the quantum determinant of $X$. How to compute $(\det_q(X))^{-1}$. Are there some references about this? Thank you very much.

Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

Math Overflow Recent Questions - Sat, 11/03/2018 - 07:56

In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra:

The space of deformations of the braided tensor structure on these particular categories is one dimensional (I believe this is a result of Drinfeld), so we get a universal family of braided deformations of U(g)-mod by varying q in U_q(g).

What is the precise reference for this result? I guess here by tensor, the answerer means equivalently monoidal?

What happend if we take out "braided"? Does there exist deformations of the monoidal category of $U(\frak{g})$-modules which are not equivalent to the cateogry of modules of $U_q(\frak{g})$, for some $q$?

A generalization of Landau's function

Math Overflow Recent Questions - Fri, 11/02/2018 - 13:13

For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$ the least common multiple of all partitions of $n$. It is the maximal order of an element in the symmetric group $S_n$.

But is there anything known if we restrict the number of summands, i.e. define $$ g(n, k) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \}, $$ Surely $g(n,1) = n$ and $g(n) = \max\{g(n,1), g(n,2), \ldots, g(n,n)\}$ and $g(n,k) = 0$ if $k > n$. Is there a recursive relation. Do you know any references where this generalization is studied?


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