Questions tagged [buildings]
The buildings tag has no usage guidance.
100
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Does every Coxeter group arise from a BN-Pair? Does $\text{PGL}_2(\Bbb{Z})$?
The question is in the title. Maybe I should explain my interest in it though. To every Coxeter group $(W,S)$ (and even more general groups) and a system of parameters $(a_s,b_s)_{s \in S}$ one can ...
3
votes
0
answers
272
views
abelian subgroups of SL(n,Q_p)
Does anyone know where I can find a classification of abelian subgroups of SL(n,Q_p) isomorphic to (Q_p)^{n-1} up to conjugacy?
Even for n=2, it would be very useful. For example, over R, there are ...
8
votes
1
answer
418
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State of the art knowledge about homology of $SL_2(k[t,t^{-1}])$
What is the current state of knowledge of the group homology of $SL_2(k[t,t^{-1}])$?
I am mostly interested in the case $k$ is algebraically closed of characteristic zero. The most recent work I am ...
2
votes
1
answer
322
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Spherical building at infinity for $SL(n, \mathbb{Q}_p)$
Is there somewhere I can read about the spherical building at infinity for $SL(n, \mathbb{Q}_p)$?
I'm looking for something with lots of explicit examples and computations. (I have books on the ...
3
votes
0
answers
128
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Extension of Tits' theorem on groups with a BN-pair of rank ≥ 3
Tits has proved that a finite simple group $G$ with a BN-pair of rank $n \ge 3$, is of Lie type. Let $B$ be the Borel subgroup and $(W,S)$ the Coxeter system. The subset lattice of the set $S$ is ...
2
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0
answers
346
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Bruhat-Tits building for $PGL_2(F)$ and repressentation theory
I am trying to understand representation theory of $GL_2(F)$ where $F$ is a finite extension of $\mathbb{Q}_p$. I am using the following paper of Colmez to understand it:
https://webusers.imj-prg.fr/~...
4
votes
1
answer
370
views
Affine building for SL(n)
Following Garrett's book, we have the usual description of affine buildings of SL(n) in terms of homothety classes of lattices. So let $F$ be a local field, $\mathcal{O}$ be its ring of integers and $\...
6
votes
2
answers
111
views
Describing the action of $^2E_6(q)$
One of the constructions of the group $^2E_6(q)$ was presented by Tits in his paper "Les «formes réelles» des groupes de type $E_6$". It is being constructed by looking at the action of $^2E_6(q)$ on ...
3
votes
1
answer
417
views
When are maximal compacts same as maximal parahorics?
Let $G$ be a reductive algebraic group over a complete non-archimedean field $k$. We know that maximal compacts are exactly the same as maximal parahorics when the Iwahori is open compact subgroup of $...
9
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2
answers
641
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What are the special parahoric subgroups in unitary groups?
Let $L$ be a $p$-adic field and let $L'/L$ be a quadratic extension. Let $U_{L'/L}(n)$ be a quasi-split unitary group of $n\times n$ matrices with entries in $L'$. I'm curious about what the special ...
5
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0
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557
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How do you understand the Moy-Prasad filtration of G_2?
Starting on page 44 of this paper of Reeder and Yu, the authors describe the first graded piece of the Moy-Prasad filtration on $G_2$ at a certain point (in this case it's $GL_2$ of the residue field),...
23
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3
answers
3k
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Is the Steinberg representation always irreducible?
Let $\mathbb{F}$ be a field. The Tits building for $\text{SL}_n(\mathbb{F})$, denoted $T_n(\mathbb{F})$, is the simplicial complex whose $k$-simplices are flags
$$0 \subsetneq V_0 \subsetneq \cdots \...
4
votes
1
answer
239
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Global bound for number of vertices in Bruhat-Tits building
Let $G$ denote a semisimple linear algebraic group over $\mathbb Q$ and let $r$ be its absolute rank.
For any prime $p$ let $v_p$ be a vertex of the Bruhat-Tits building of $G({\mathbb Q}_p)$ and let $...
0
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0
answers
151
views
Buildings for Affine groups
Let $G$ denote one of the classical groups over a finite field. Is there a natural way to associate a building to the affine group $V\rtimes G$, and an analog of the Solomon-Tits theorem?
1
vote
1
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Pairs of rays in euclidean buildings
In section 4.1.3 of Kleiner and Leeb's paper on symmetric spaces and euclidean buildings, there's a result about pairs of rays from the same point initially spanning a flat triangle (or being ...
3
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1
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Is a cocompact CAT(0) periodic?
Let $X$ be a CAT(0) space and $G$ its group of isometries. Then $X$ is said to be cocompact, if there exists a compact set $K\subset X$ with $X=G.K$. The space $X$ is called periodic, if there exists ...
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2
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Understanding how to construct Bruhat-Tits buildings for non-split groups by Galois descent
Is there any way to get on top of the procedure for constructing Bruhat-Tits buildings for non-split groups over a non-archimedean local field $k$, by Galois descent, other than reading both the ...
4
votes
1
answer
548
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Fixed points in the Bruhat-Tits building
Let $G$ be a connected reductive group over a complete discrete valuation field with perfect residue field (or just a non-arch local field). Let $\mathcal{B}$ be its reduced Bruhat-Tits building, and $...
5
votes
1
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Arithmetic quotients of Bruhat-Tits buildings for groups over local fields of positive characteristic
I have been led to believe that there is a result giving a description of the quotient of a Bruhat-Tits building $\Delta(G,k)$, for a semisimple algebraic group $G$ over a non-archimedean local field ...
4
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0
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Bruhat Tits buiding to visualize closed points of affine flag varieties?
In his survey "affine springer fibers and affine Deligne-Lusztig varieties", Goertz gives us a tutorial session on how to use Bruhat Tits buildings to visualize subsets of affine Grassmannians or of ...
6
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1
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When are toral orbits in buildings the difference of fixed-sets?
Let $L$ be a $p$-adic field, let $G$ be a reductive group over $L$ (I'm even okay assuming semisimplicity for now). Let $T$ be a maximal torus of $G$. Let $B$ be the building for $G(L)$. (Edit 1: "...
6
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1
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509
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When do two lattices have the same stabilizer in the diagonal torus?
This is moved from MSE, where I asked and didn't receive an answer (see https://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer)
Let $T$ be the diagonal torus ...
8
votes
1
answer
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Embedding Euclidean buildings into products of trees
A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.)
Question: Is it true ...
1
vote
2
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189
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Question on affine buildings
Let $X$ be an affine building. Assume that $X$ is periodic, by which I mean that there exists a covering $X\to F$ of a finite simplicial complex. Let $\Gamma$ denote the group of deck transformations, ...
3
votes
1
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247
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algebraic groups over non-archimedean local fields acting on buildings
I was wondering could anyone tell me a reference for the fact that an absolutely quasi-simple algebraic group over a non-archimedean local field which is centreless and non-compact acts faithfully and ...
2
votes
0
answers
246
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(Co)Homology of groups vs. Lie algebras: polynomial rings
For Lie groups (or algebraic groups over fields) there is a strong relation between the cohomology of the group and the cohomology of its Lie algebra. Some MO-question where this is discussed can be ...
6
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0
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222
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Parshin's buildings for higher local fields
What is the status of the theory of buildings for higher local fields?
I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over two-...
9
votes
2
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649
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$G_\mathbb{Z}$-homotopy type of rational Tits building $\Delta_{G, \mathbb{Q}}$
Take $G$ to be a standard semisimple algebraic $\mathbb{Q}$-group, e.g. $Sp_{2g}$ or $SO(h)$ for $h$ a nondegenerate quadratic form over $\mathbb{Q}$. The arithmetic group $\Gamma=G_{\mathbb{Z}}$ has ...
4
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0
answers
175
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Rational cohomology of $S$-arithmetic groups over function fields and Gauss-Bonnet
I have a question on the ranks of rational cohomology groups of
$S$-arithmetic groups over function fields. To fix the situation, $G$
is a simple Chevalley group of rank $r$, $k=\mathbb{F}_q$ a finite
...
4
votes
2
answers
334
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Does a spherical building embed in a building of type $A_n$?
I'm interested in the question in the title:
Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$?
By embedding I mean an isometric embedding with respect ...
6
votes
1
answer
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Clarification about Tits' article in the Corvallis
I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ (...
8
votes
1
answer
366
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spherical buildings for non-split groups
I am looking for references to explicit descriptions of Tits buildings for semisimple (classical) Lie groups via language of incidence geometry. Such descriptions are well-documented in the case of ...
5
votes
1
answer
223
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Density/Thickness of rank 3 spherical buildings
I am trying to study (finite) spherical buildings from a very combinatorial point of view : Every rank 3 spherical building is a finite simplicial complex of dimension 3, so one can define its density ...
7
votes
1
answer
480
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When is a Moy-Prasad filtration subgroup the stabilizer of a subset of the building (up to center)?
Let $G$ be a connected, simply connected, semi-simple algebraic group defined and split over a local non-arch field $k$ with integer ring $R$. Let $B$ be the corresponding reduced building. Fix an ...
11
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3
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What is a (generalized) BN-pair?
Let us consider $GL_n(K)$ over a local field $K$. It has standard subgroups $N$ and $B$. $B$ is Iwahori subgroup, $N$ consists of monomial matrices. The pair comes close to a romantic ending, i.e. ...
1
vote
1
answer
272
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Heights in reductive groups
Let $G$ be a reductive group over a local non-archimedean field $F$, and let $B$ a Borel subgroup. For my purposes, the case $G = GL_2(\mathbb{Q}_p)$ will be sufficient with $B$ upper triangular ...
1
vote
2
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211
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What is this measure on the affine Weyl group?
Let $G$ be $SL(n, F)$ for a non-archimedean local field $F$ with Iwahori subgroup $I$. Let $\mu$ be the Haar measure of $G$.
What are the properties of the function $w\mapsto \mu(IwI)/\mu(I)$ for $w$...
8
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2
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Conjugation in GL(n) (p-adic setting)
In $GL(n, \mathbb{Q}_p)$, what are the orbits under conjugation of $GL(n, \mathbb{Z}_p)$?
2
votes
2
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861
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Parabolic subgroups and BN-pairs
We note $G$ a connected algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup containing $T$. We put also, $N=N_{G}(T)$ the normalizer of $T$ in $G$. We know that $(B,N)$ is a BN-pair of $...
1
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1
answer
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Algorithm for the cell multiplication rule for GL(n,F)
Consider $F$ a non archimedean field and let $o$ be its ring of integer
Let $B$ be the Iwahori subgroup of $GL_n(F)$ (resp. $GL_n(o)$) and let $N$ be the normalizer of the diagonal matrices (...
2
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2
answers
568
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What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building?
$F$ is a non archimedean field here. To be more precise, I would actually prefer a set of representative in $B(F)$ for the discrete space $B(F) / B(o)Z(F)$?
This can be phrased also as question about ...
11
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2
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Spherical building of an exceptional group of Lie type
I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...
9
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2
answers
865
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Subexpressions of reduced words in Coxeter groups
Let $\underline{w} = [s_1, s_2, \dots ,s_n]$ be a reduced expression in a Coxeter group $W$. Given $x$ in $W$ one can consider the set $\Pi(\underline{w},x)$ consisting of all subexpressions of $\...
21
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6
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8k
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Any good reference for Tits Building?
For beginers, any suggestions?
3
votes
3
answers
480
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Is this the CAT(0) metric on an affine building?
Let $R$ be a discrete valuation ring qith quotient field $Q$ and let $t\in R$ be a generator of the unique maximal ideal in $R$. Let $V$ be a finite-dimensional $Q$-vector space. Then one can consider ...
9
votes
1
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On which space does $GL_n(F_p[X])$ act nicely?
The group $GL_n(\mathbb{Z})$ acts properly and isometrically on the space of homothety classes of scalar products on $\mathbb{R}^n$. This is a Riemannian manifold of nonpositive sectional curvature.
...
9
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1
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Proof of an 'easy' exercise in a book of Tits
In 'Buildings and Finite $BN$-Pairs', Jacques Tits gives the following statement which is left as an easy exercise.
Let $G_1,G_2,G_3$ be three subgroups of a group $G$. Then the following conditions ...
5
votes
2
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Geometric interpretation of $BN$-pairs
My question is relative to a geometric interpretation of the $BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (Nonspherical spheres).
$[...
14
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1
answer
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Distance to an apartment of the affine building of GL(N)
Here $F$ is a locally compact non-archimedean non-discrete field.
Let $X$ be the reduced (affine) Bruhat-Tits building of ${\rm GL}(n,F)$. Fix a maximal split torus $T$. Let $B$ be a Borel subgroup ...
10
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3
answers
1k
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Looking for figure of part of an A2 affine building
Somewhere, I don't remember where, I saw a beautiful 3D figure of part a CAT(0) simplicial complex. I am thinking and hoping that this was some finite piece of an affine building of type A2, ...