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Let $R$ be a reduced root system, $W$ the associated Weyl group, and $w_0 \in W$ the longest element of $W$. In general $w_0$ admits more than one reduced decomposition into a product of reflections, a number which we denote by $d_R$. Where can one find a list of values of $d_R$ for low-dimensional root systems?

For example are the explicit values of $d_R$ known for the exceptional root systems?

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    $\begingroup$ For Types A and B(=C) there are product formulas for these numbers: see the famous paper math.mit.edu/~rstan/pubs/pubfiles/56.pdf and math.stackexchange.com/questions/2271510/…. I'm pretty sure for Type D there is not a product formula (as Stanley mentions there is a big prime, 193, in the factorization of the number of reduced words of the longest word in Type $D_4$). As for exceptionals I don't know of a list but this is in principle something a computer can do. $\endgroup$
    – Sam Hopkins
    Commented Aug 28, 2020 at 16:36
  • $\begingroup$ In the linked stackexchange webpage the B-series formula in the answer gives, for low values of $n$, a product with negative limits. Is this an error or is this an example of some convention I am not familar with? For example, what is the value for $B_2$? $\endgroup$
    – Bas Winkelman
    Commented Aug 28, 2020 at 17:07
  • $\begingroup$ Maybe the formula Zach wrote is not quite right. For $B_2$ the answer should be 2. It is the same as the number of linear extensions of the root poset (poset of positive roots whereby $\alpha \leq \beta$ if $\beta-\alpha$ is a nonnegative sum of simple roots). This poset is the same as the shifted trapezoid shape $(2n-1,2n-3,...,1)$ poset. This number also happens to be the same as the number of SYTs of $n\times n$ square shape. $\endgroup$
    – Sam Hopkins
    Commented Aug 28, 2020 at 17:14
  • $\begingroup$ You can see Corollary 5.2 of the paper of Stanley linked above for another way of writing the product formula for the Type B # of reduced decompositions (he lists it as a conjecture but it has been proven). $\endgroup$
    – Sam Hopkins
    Commented Aug 28, 2020 at 17:16
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    $\begingroup$ The basic thing that's going on here is that there's an Edelman-Greene style bijection in Types A and B (and also the non-Weyl types $I_2(m)$ and $H_3$- sometimes these types are called the 'coincidental types'). The other types don't have such a bijection. $\endgroup$
    – Sam Hopkins
    Commented Aug 28, 2020 at 17:18

2 Answers 2

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This is easy to do in SageMath. E.g. the following code

G = WeylGroup("F4")
w = G.long_element_hardcoded()
print(w)
rw = w.reduced_words() 
len(rw)

outputs 2144892. If you want to look at some of these reduced words just examine the list rw. To create a list for classical types of different rank do

res = {}
for n in range(2,5):
    G = WeylGroup(["A", n])
    w = G.long_element_hardcoded()
    print("Calculating rank ", n)
    res[n] = len(w.reduced_words())
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    $\begingroup$ Note 2144892 = 2^2 x 3 x 47 x 3803, again suggesting lack of product formula. $\endgroup$
    – Sam Hopkins
    Commented Aug 28, 2020 at 19:34
  • $\begingroup$ I created findstat.org/StatisticsDatabase/St001585. If you have enough power to do E6 that would be great! $\endgroup$
    – Martin Rubey
    Commented Aug 28, 2020 at 19:44
  • $\begingroup$ @MartinRubey I will try some computer in Prague's Charles University to which I have remote access. But I am pessimistic. This algorithm actually generates and stores all the reduced words so it eats all available memory rather quickly. $\endgroup$
    – Vít Tuček
    Commented Aug 28, 2020 at 19:54
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    $\begingroup$ Counting reduced words is the same as counting directed paths between two vertices in a certain directed graph (the weak order graph), so it's possible you could write code by hand that does this faster than what's in Sage. $\endgroup$
    – Sam Hopkins
    Commented Aug 28, 2020 at 19:59
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    $\begingroup$ @SamHopkins, indeed, this is much faster! I added corresponding code to the findstat entry. $D_5$ is now easy. Unfortunately, FindStat has a limitation on the size of statistic values, so anything beyond $A_6$ is too large. $\endgroup$
    – Martin Rubey
    Commented Aug 28, 2020 at 20:20
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Around the time Fomin and I wrote this paper, Tao Kai Lam applied the technique to type $D_n$. It emerged that it was "natural" to weight a reduced decomposition $\rho$ by $2^{d(\rho)}$, where $d(\rho)$ is the number of simple reflections in $\rho$ that correspond to the $n-2$ "nonbranch nodes" in the Coxeter diagram for $D_n$. Using this weighting, there is a nice product formula for the number of weighted reduced decompositions of the longest element, which I unfortunately have forgotten. I hope someone can redo this work.

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  • $\begingroup$ Is this weighted sum related to linear extensions in some way? $\endgroup$
    – Sam Hopkins
    Commented Aug 28, 2020 at 19:09
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    $\begingroup$ @SamHopkins: I have a hazy recollection that it is a power of 2 times the number of SYT of the shifted shape $(2n-2,2n-4,\dots,4,2)$ (for which there is a nice product formula). $\endgroup$
    – Richard Stanley
    Commented Aug 28, 2020 at 19:41
  • $\begingroup$ That would make a lot of sense! In conservancy.umn.edu/bitstream/handle/11299/159973/… Williams calls the poset you're talking about the "flattened root poset" of $D_n$. $\endgroup$
    – Sam Hopkins
    Commented Aug 28, 2020 at 19:44
  • $\begingroup$ Actually, the result you mention might follow from this paper of Billey and Haiman: math.berkeley.edu/~mhaiman/ftp/schubert/schubert.pdf (see also the discussion on pg. 46 of the thesis of Williams linked above) $\endgroup$
    – Sam Hopkins
    Commented Aug 28, 2020 at 19:46
  • $\begingroup$ Name of this paper: Schubert polynomials and the nilCoxeter algebra. $\endgroup$
    – LSpice
    Commented Nov 23, 2023 at 17:27

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