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Let $P$ be a finite poset which is a lattice with $0,1 \in P$. An atom in $P$ is an upper cover of $0$ and a coatom is a lower cover of $1$. $P$ is atomic if every element is a join of atoms and coat …

This is more of an expansion of Sam's comments, but too long for a comment itself: As pointed out by Sam in that Theorem 4.2.2.2, tilting modules of the linear type $A_n$ quiver are in natural corres …

Let $E_n(z)$ be the Eulerian polynomial $$E_n(z) = \sum_{\tau \in \mathfrak{S}_n} z^{\operatorname{des}(\tau)}$$ where $\mathfrak{S}_n$ denotes the set of all permutations of $\{1,\ldots,n\}$ and $\op …

You want to print a deck of cards of the following type: Each card shows $k$ items out of $n$ different items such that any two cards in the deck share exactly one item. Question: What is the bigg …

If I am not mistaken, the space $Alt$ has a basis given by $\{ \Delta_M(\mathbf{x},\mathbf{y}) \mid M \subset \mathbb{N}\times\mathbb{N} \text{ finite}\}$. Here, $\Delta_M(\mathbf{x},\mathbf{y}) = \d …

Let $V$ be a complex vector space of finite dimension $n$ and let $W$ be a finite unitary reflection group. This is, $W$ is a subgroup of $GL(V)$ generated by reflections, i.e., elements $r \in GL(V)$ …

A complete combinatorial proof using Allen's comment: Let $(W,S)$ be a Coxeter system, and let $Dem(T) \in W$ be the Demazure product or greedy product of a word $T$ in $S$. Claim: $Dem(T)$ is the u …

As a follow-up of Allen's question Coxeter exchanges in non-reduced words, I wonder whether it is known that the Demazure product is well-defined in Artin groups. This is: Let $(W,S)$ be a Coxeter sy …

(in contrary to what I thought first,) here is a proof that the "exchange condition" holds in the following sense. It is based on the root configuration in http://arxiv.org/abs/1111.3349 [1]. Let $(W …

These are several comments rather than an answer: As you write, let $(Q,w,S)$ denote all subwords of $Q$ whose greedy product is $w$ with positions $S$ skipped. I moreover write $(Q,w)$ for w in the …

(This is more of a longer comment without a proper answer to the question.) I assume you mean "An embedding $(W,S) \hookrightarrow (W',S')$ is an injective map from $S$ to the set $T' = \{ w s w^{-1} …

For future references: as we have discussed in this question, this also follows from the "zeta map" sending the bistatistic (area,bounce) to the bistatistic (dinv,area). For another definition and fur …

Eduard Looijenga provided a scanned version of the letter, which now can be found at http://homepage.univie.ac.at/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf (outdated) http://homepage.r …

Hello, I wonder if anyone has a copy of Deligne's letter to Looijenga from 1974 mentioned as reference [26] in Bessis' paper Finite complex reflection arrangements are $K(\pi,1)$ from 2006, see http: …

Symmetric statistics on Dyck paths: The following statistics have a symmetric joint distribution on Dyck paths: (area,bounce), see here (area,dinv), see here (number of returns, length of last desc …