19
votes
Have you seen my matroid?
One can use Whitney's theorem to show that the characteristic polynomial is
$$ \sum_{i=0}^k{n\choose i}q^{k-i}(q-2)^{n-i} +
\sum_{i=k+1}^n{n\choose i}(q-2)^{n-i}. $$
I doubt that this can be ...
- 47.9k
19
votes
Can $E_8$ be enlarged?
No.
Let $G$ be a finite subgroup of $GL_n(\mathbb R)$ containing $W(E_8)$ as a subgroup. Because $G$ is compact, $G$ must preserve a symmetric positive definite form on $\mathbb R^8$. Since $W(E_8)$ ...
- 129k
14
votes
Why do combinatorial abstractions of geometric objects behave so well?
Perhaps this, for now, is more an issue of perspective. Yes, for matroids, spheres and Coxeter groups the realizable cases were known before using results in algebraic geometry, but this is natural as ...
- 674
14
votes
Accepted
Have you seen my matroid?
Let $U$ be the uniform matroid of rank $k$ on $n$. Since $U$ is orientable one can consider the Lawrence oriented matroid $\Lambda(U)$ associated with any orientation of $U$ (the Lawrence construction ...
- 974
12
votes
Accepted
Swapping non-commuting generators in Coxeter group
The answer is no. The Deletion Condition says that any expression in the generators of a Coxeter group contains a reduced expression for the same element as a subexpression. Since $a$ and $b$ don't ...
- 1,522
12
votes
Accepted
One element commutation classes of reduced decompositions of the longest element of the Weyl group
The reduced words that are in their own commutation classes are:
$s = [123\cdots(n-2)(n-1)(n-2)\cdots321][23\cdots(n-3)(n-2)(n-3)\cdots32][3\cdots(n-4)(n-3)(n-4)\cdots3]\cdots$
the reversal of $s$ (...
- 136
11
votes
Accepted
Instructions for using Coxeter 3.0 software
If you type "help" immediately on entering the program, you'll get a fairly long and useful introductory message. At whatever level you are, you are supposed to be able to type "help," and then the ...
- 521
11
votes
Accepted
Subgroups of RAAGs vs. subgroups of RACGs
The fundamental group of the (non-orientable) closed surface of Euler characteristic -1 provides a counterexample.
On the one hand, it’s a subgroup of index 4 in the reflection group on the right-...
- 23.5k
10
votes
Why do combinatorial abstractions of geometric objects behave so well?
As Uri Bader remarked one has to be careful with the term "combinatorial abstraction". In the cases mentioned by Sam and in other cases the geometric objects are certain algebraic varieties but the ...
- 24.2k
10
votes
Accepted
A duality result for Coxeter groups
Yes. By Chevalley-Shepard-Todd, $S(V)^G$ and $S(V)^H$ are polynomial rings. Let $S(V)^G=\mathbb{R}[g_1, \ldots, g_n]$ and $S(V)^H = \mathbb{R}[h_1,\ldots, h_n]$ where the $g_i$ and $h_i$ are ...
- 148k
10
votes
Subgroups of $W(E_8)$
One approach is to calculate the orbits of $W(A_8)$ on $W(E_8) / W(A_8)$.
I claim these orbits have sizes $1, 1, 84, 84, 560, 560, 630$.
Given this claim, it's straightforward to check. For any ...
- 129k
10
votes
Accepted
When are groups generated by reflections in a triangle discrete?
The Euclidean case is an easy case-by-case analysis.
The hyperbolic case was resolved by Anna Felikson, encapsulated in this figure from her paper:
A triangle with mirrored sides may be regarded as a ...
- 65.5k
9
votes
One element commutation classes of reduced decompositions of the longest element of the Weyl group
I believe that for $n \geq 4$ there will be exactly $4$ such reduced words. One such word, call it $R_n$ can be constructed by starting with $s_{n-1}s_{n-2} \cdots s_2s_1s_2 \cdots s_{n-2}s_{n-1}$ and ...
- 2,673
8
votes
Can you prove Givental's conjecture on wavefronts and the icosahedron?
In http://www.sciencedirect.com/science/article/pii/S0167278998900057 (Remarks on quasicrystallic symmetries) Arnold so describes the idea of Shcherbak's proof:
Now the proof of this theorem ...
- 16.2k
8
votes
Why do combinatorial abstractions of geometric objects behave so well?
I'm not exactly addressing your question about combinatorial abstractions of geometric objects, but you seem to be taking Lie theory as a given natural geometric arena.
On the contrary, the ...
- 2,880
8
votes
Accepted
Does every Coxeter group arise from a BN-Pair? Does $\text{PGL}_2(\Bbb{Z})$?
$PGL_2(\mathbf{Z})$ is a Coxeter group with presentation $\langle s_1,s_2,s_3 \mid s_i^2=(s_1s_2)^3=(s_1s_3)^2=1 \rangle$, i.e. with labels $\{2,3,\infty\}$. To each Coxeter group $W$ with labels in $\...
- 4,148
8
votes
Accepted
Vanishing of certain coefficients coming from Coxeter groups
The answer is yes. I claim there are at most $2^n$ elements of $W$ for which $\lambda(w) \neq 0$. Specifically, let $\vee$ be the join in weak order, which is a semi-lattice. If $\{ s_1, s_2, \ldots, ...
- 148k
8
votes
Accepted
Reduced expressions for reflections in a Coxeter group
Depending on what you mean by a "good way", maybe there is and maybe there isn't. If you want to do this all in terms of the combinatorics of reduced words, probably the following is the best you can ...
- 1,806
8
votes
Accepted
Number of paths in the Bruhat order in the symmetric group
$\ell(v)!$ is of course even if $\ell(v)>1$, so the statement is really that $N_v$ is even for $\ell(v)>1$. We find a fixed-point free involution on the set of such Bruhat paths. Suppose that $...
- 38.7k
8
votes
Which reflection groups can be enlarged?
$A_1^*$ is maximal.
$A_2 = I_3$ and $A_2^* = I_6 = G_2$ (for the first one, consider the Dynkin diagram, for the second one, any group containing it as an index $2$ subgroup must normalize it, hence ...
- 129k
7
votes
Accepted
Quotient of Coxeter complex in terms of double cosets?
I think this is the same issue that I noticed later myself, and addressed in Remark 2.13 on page 228 of the paper that I wrote with Eric Babson: "Coxeter-like complexes" (Discrete Mathematics and ...
- 186
7
votes
Accepted
Why is the root poset is graded by height?
Let $(\cdot,\cdot)$ be a positve definite Weyl group invariant product on $\mathbb{R}\Phi$. Let $\beta$ and $\gamma$ be positive roots with $\beta\leq \gamma$ and $h(\gamma)-h(\beta)\geq 2$. Let $v=\...
- 8,586
7
votes
Accepted
Number of pairs of permutation in $S_n$ whose $\mu$-coefficient (of their Kazhdan Lusztig polynomial) is non-zero
When I calculated these numbers using Marc van Leeuwen and Fokko du Cloux's software atlas, I got
S_3: 8
S_4: 60
S_5: 482
S_6: 4268
S_7: 41934
S_8: 457782
(I could easily have made some silly ...
- 521
7
votes
Accepted
Is the number of commutation classes of reduced words of the longest element of $S_n$ even for $n\geq 3$?
On the linked OEIS entry I find the following:
Also the number of mappings X:{{1..n} choose 3}->{+,-} such that for any four indices a < b < c < d, the sequence X(a,b,c), X(a,b,d), X(a,c,d)...
- 38.7k
7
votes
Accepted
Lattice structure in the root poset
The root poset for $\tilde{A_2}$ is shown in Figure 4.5 in the same reference and copied below. One can check that the elements labelled $112$ and $221$ have both $100$ and $010$ as maximal common ...
- 2,673
7
votes
Accepted
Product of two reflections lying in a parabolic subgroup of a Coxeter group
$\DeclareMathOperator\Im{Im}\DeclareMathOperator\Fix{Fix}$As suggested by Sam, I am posting this as an answer.
As mentioned in the comments, the above question can be solved using a result of ...
- 484
6
votes
Are there Hamilton paths in Cayley graphs of Coxeter groups?
The answer to both questions is yes. See the paper of Conway, Sloane and Wilks called "Gray codes for reflection groups":
http://link.springer.com/article/10.1007/BF01788686
- 3,681
6
votes
Accepted
Injection from Artin monoids to Coxeter groups
This is a general fact on Artin-Tits groups attached to Coxeter groups. Let $B(W)^+$ be the Artin-Tits monoid of the Coxeter group $W$. Then there is a canonical surjection $B(W)^+\rightarrow W$ which ...
- 484
6
votes
Reference request: Reduced reflection length in Coxeter groups
After contacting the authors of the paper mentioned above, I want to share the information they gave me with anyone who might ever stumble upon this question. For that, define an ordering on $G$ by ...
- 799
6
votes
Accepted
Elements of Coxeter group whose simple reflections pairwise commute
If I see it correctly, there is not much going on in the set $X(G)$ from the viewpoint of Coxeter systems: Let $S$ be the simple system and let $G$ be the Coxeter graph of $(W,S)$ with vertex set $S$. ...
- 3,211
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