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comment List is a monad, but is it a comonad with these natural transformations?
That's a nice comonad! Interestingly, you can obtain another comonad on the same functor $L^+$ by using cyclic shifts instead of tails. I am wondering what its coalgebras are and whether it might shed some light on necklaces and the Burrows-Wheeler and Gessel-Reutenauer transforms.
Apr
29
revised proof that ${\rm SL}_n (R)=E_n(R)$ in a local ring?
added 8 characters in body
Apr
28
answered Unique factorization of posets
Apr
28
comment Unique factorization of posets
In full generality, your question has a negative answer: See Tadasi Nakayama and Junji Hashimoto, On a problem of G. Birkhoff, Proc. Amer. Math. Soc. 1 (1950), pp. 141--142, ams.org/journals/proc/1950-001-02/S0002-9939-1950-0035279-X .
Apr
28
comment Unique factorization of posets
Actually, a positive answer to your question in the case where $P$ is connected is claimed in: Junji Hashimoto, On Direct Product Decomposition of Partially Ordered Sets, Annals of Mathematics, Second Series, Vol. 54, No. 2 (Sep., 1951), pp. 315--318, sci-hub.io/10.2307/1969532# .
Apr
28
comment Unique factorization of posets
If the posets in question all have a global minimum and a global maximum, then this follows from Lemma 6.1 in: William R. Schmitt, Incidence Hopf algebras, home.gwu.edu/~wschmitt/papers/iha.pdf .
Apr
19
comment Collecting proofs that finite multiplicative subgroups of fields are cyclic
(BTW @PeteL.Clark: Care to check that you don't have several equations labeled identically (logfile warning: "LaTeX Warning: Label `[name of your label]' multiply defined.") in your tex source? On page 8, "Theorem 140" and "Theorem 142" are mentioned, but probably refer to Theorems 7 and 8. This is a particularly easy mistake to make when you are including many texs in a single file.)
Apr
19
comment Collecting proofs that finite multiplicative subgroups of fields are cyclic
For the record: The theorems and proofs mentioned in this answer are probably in Section B.2 of math.uga.edu/~pete/4400FULL.pdf nowadays.
Apr
16
comment Examples of combinatorial bijections found by considering functors
I remember Viviane Pons telling me once that she found a bijection between two objects (in Catalan combinatorics) by narrowing down the search space by requiring the bijection to respect some statistics (after checking with Sage that said statistics were equidistributed). Sadly, I don't remember the details.
Apr
16
revised Tableaux with limited rows and complementary skew shapes
partially read; the "all" can be misunderstood as saying that all elements of [n] are in there
Apr
15
comment Demazure product in Coxeter and Artin groups
(Note that the more general version of Lemma 1 directly follows from Lusztig's Corollary 2.5 (a), whereas Björner-Brenti's Proposition 2.2.7 only gives the weaker version; things like this make Lusztig my go-to place for lemmas about Coxeter groups.) Other than this, thanks for the nice proof! It sort-of proves the claim of mathoverflow.net/questions/81539 as well, I believe.
Apr
15
comment Demazure product in Coxeter and Artin groups
Actually, I think that when you say you're using Lemma 1, you actually use the more general version where $s$ is allowed to be a right descent of $u$ and where the condition $u < w$ is replaced by $u \leq w$ (because you apply it to $u = \prod \left(Q \setminus t_m\right)$ and $w = Dem(T')$). This version easily follows from your Lemma 1 (indeed, it is easy to prove $us \leq w$ both in the case $u=w$ and in the case $us<u$), but unless you replace Lemma 1 by this version I think you cannot claim that "we are in the situation of Lemma 1".
Apr
15
revised Demazure product in Coxeter and Artin groups
define right ascents too
Apr
15
comment Demazure product in Coxeter and Artin groups
Note: Your Lemma 1 also follows from Corollary 2.5 (a) in Lusztig's arXiv:math/0208154v2, applied to $y = u^{-1}$ and $z = w^{-1}$. Notice that the requirement that $s$ not be a right descent of $u$ is not needed.
Apr
15
revised Demazure product in Coxeter and Artin groups
1) fix inequality (strict < fails for u = ws); 2) define right descents (not sure if everyone is on the same page on this notion); 3) the set, not the family (else, "unique" would be false, I believe)
Apr
15
revised Demazure product in Coxeter and Artin groups
added 134 characters in body
Apr
13
comment Schubert varieties and Young diagrams
The second part asks you to prove that none of the conditions can be omitted, right? Your restatement doesn't make that much sense to me.
Apr
10
comment The augmentation filtration on a group ring
The counit is uniquely determined by the coproduct; thus, yes.
Apr
10
comment The augmentation filtration on a group ring
Isn't $I/I^n $ still the kernel of the counit?
Apr
9
comment A question on symmetric functions
I forgot to say: $\mu^t$ means the transpose (aka conjugate) of the partition $\mu$. Generally, the notation is that used in arXiv:1409.8356v3, and the well-known result I am using is part of the proof of Proposition 2.17. Sorry for the messy comments where a clean answer would have been far more appropriate; my spare time is currently not in abundance :(