50,178 research outputs found
Properties of Non-symmetric Macdonald Polynomials at q = 1 and q = 0
No full textAbstract
We examine the non-symmetric Macdonald polynomials
E
λ
at
q
=
1
, as well as the more general permuted-basement Macdonald polynomials. When
q
=
1
, we show that
E
λ
(
x
;
1
,
t
)
is symmetric and independent of t whenever
λ
is a partition. Furthermore, we show that, in general
λ
, this expression factors into a symmetric and a non-symmetric part, where the symmetric part is independent of t, and the non-symmetric part only depends on
x
, t, and the relative order of the entries in
λ
. We also examine the case
q
=
0
, which gives rise to the so-called permuted-basement t-atoms. We prove expansion properties of these t-atoms, and, as a corollary, prove that Demazure characters (key polynomials) expand positively into permuted-basement atoms. This complements the result that permuted-basement atoms are atom-positive. Finally, we show that the product of a permuted-basement atom and a Schur polynomial is again positive in the same permuted-basement atom basis. Haglund, Luoto, Mason, and van Willigenburg previously proved this property for the identity basement and the reverse identity basement, so our result can be seen as an interpolation (in the Bruhat order) between these two results. The common theme in this project is the application of basement-permuting operators as well as combinatorics on fillings, by applying results in a previous article by Per Alexandersson
Linear differential operators and their Hutchinson invariant sets [Elektronisk resurs]
No full textIn this licentiate thesis we consider problems related to what we call Hutchinson invariance, which is a form of invariance for sets in the complex plane or the Riemann sphere with respect to the action of special differential operators.In the introductory chapter, we provide a background on Hutchinson invariance, explain how it relates to other problems in dynamical systems and why it is an interesting subject of study. In particular, we relate it to the Pólya-Schur theory, rational vector fields as well as iterations of rational functions and algebraic correspondences.Paper I is joint with my principal and secondary supervisors, Boris Shapiro and Per Alexandersson. It studies what we call continuous Hutchinson invariance for first order differential operators, which is a special form of invariance for sets in the complex plane. We investigate a variety of properties of these invariant sets. For instance, we describe when there exists a minimal under inclusion continuously Hutchinson invariant set, when it is compact, when it coincides with the whole complex plane and when it equals a line or line-segment.Paper II is entirely written by myself. It studies what we call merely Hutchinson invariance, which is a concept that is closely related to that studied in Paper I. In this second paper, we among other things find that there exists a minimal under inclusion Hutchinson invariant set for a large class of linear operators, and that in this case, this set is bounded and perfect.</p
LLT polynomials, elementary symmetric functions and melting lollipops
No full textWe conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Shareshian-Wachs conjecture previously studied by Panova and the author in 2018. We show that the conjecture for unicellular LLT polynomials implies a similar formula for vertical-strip LLT polynomials. We prove positivity in the elementary symmetric basis for the class of graphs called melting lollipops previously considered by Huh, Nam and Yoo. This is done by proving a curious relationship between a generalization of charge and orientations of unit-interval graphs. We also provide short bijective proofs of Lee's three-term recurrences for unicellular LLT polynomials, and we show that these recurrences are enough to generate all unicellular LLT polynomials associated with abelian area sequences.</p
The cyclic sieving phenomenon on circular Dyck paths
No full textWe give a q-enumeration of circular Dyck paths, which is a superset of the classical Dyck paths enumerated by the Catalan numbers. These objects have recently been studied by Alexandersson and Panova. Furthermore, we show that this q-analogue exhibits the cyclic sieving phenomenon under a natural action of the cyclic group. The enumeration and cyclic sieving is generalized to Mobius paths. We also discuss properties of a generalization of cyclic sieving, which we call subset cyclic sieving, and introduce the notion of Lyndon-like cyclic sieving that concerns special recursive properties of combinatorial objects exhibiting the cyclic sieving phenomenon.</p
Studies on colonoscopy and inflammatory conditions of the colon
No full textBackground: Colonoscopy is a key investigative tool in inflammatory bowel disease and diverticular disease, but elements of the pathogenesis, symptomatology and safety of procedures in these two conditions remain unknown. The best surveillance method for colorectal cancer in inflammatory bowel disease (IBD) is debated. Chromoendoscopy (CE) is recommended, but whether CE is the best method in high-definition (HD) colonoscopy is unclear. IBD causes inflammation and sometimes fibrosis in the colon that might lead to a more fragile colon. Whether this is linked to a higher rate of adverse events in colonoscopy is unknown. The cause of diverticular disease is mostly unknown. One plausible cause is difference in microbiota composition, but studies are limited. Pain and other symptoms are not well documented in uncomplicated diverticulosis and studies are scarce.Methods and Main results: Paper I: A randomized study of IBD-patients undergoing surveillance colonoscopy comparing HD-CE with random biopsies vs HD-White Light Endoscopy (WLE) with random biopsies. Three hundred and five patients were included (n = 152, HD-CE; n = 153, HD-WLE). Dysplastic lesions were found in 17 individuals with HDCE and 7 individuals with HD-WLE (p = 0.032). The number of dysplastic lesions found for every 10 minutes of withdrawal time was 0.066 with HD-CE and 0.027 with HD-WLE (p = 0.056). Paper II: A nationwide, population-based register study in Sweden examining all colonoscopies between 2003 and 2019. The association between IBD and other risk factors with increased bleeding or perforation rates in colonoscopy was assessed. All incidents of bleeding and perforation within 30 days after each colonoscopy were recorded. Bleeding (0.19%) and perforation (0.11%) were found to be rare. Bleeding (OR 0.66, p Conclusions: These data indicate that in IBD patients, HD-CE is superior to HD-WLE endoscopy in finding dysplastic lesions per colonoscopy as well as more efficient per 10 minutes of withdrawal time. We recommend the use of HD-CE in surveillance of IBD patients. Individuals with IBD did not experience more adverse events compared with individuals without IBD. This implies that no special considerations need to be taken when performing colonoscopy on individuals with IBD. There was no sign of difference in microbiota composition when comparing individuals with and without diverticulosis. Individuals that later developed acute diverticulitis had a higher abundance of genus Comamonas but the significance of this is unclear and may suggest a limited role for microbiota in the pathogenesis of diverticular disease. Diverticulosis was not associated with more abdominal pain or more LLQ abdominal pain when compared to individuals without diverticulosis. Abdominal pain was strongly associated with a diagnosis of IBS.List of scientific papersI. HD chromoendoscopy in IBD surveillance. Alexandersson B, Hamad Y, Andreasson A, Rubio CA, Ando Y, Tanaka K, et al. High-Definition Chromoendoscopy Superior to High-Definition White-Light Endoscopy in Surveillance of Inflammatory Bowel Diseases in a Randomized Trial. Clin Gastroenterol Hepatol. 2020;18(9):2101-7. https://doi.org/10.1016/j.cgh.2020.04.049 II. Inflammatory bowel disease and colonoscopy. Alexandersson B, Andreasson A, Hedin C, Broms G, Schmidt PT, Forsberg A. Inflammatory bowel disease is not linked to a higher rate of adverse events in colonoscopy — a nationwide population-based study in Sweden. Journal of Crohn’s & Colitis. 2023. https://doi.org/10.1093/ecco-jcc/jjad114 III. Gut microbiota in diverticulosis. Alexandersson B, Hugerth LW, Hedin C, Andreasson A, Schmidt PT et al. Diverticulosis is not associated with altered gut microbiota nor is it predictive of future diverticulitis, a population-based colonoscopy study. Scandinavian Journal of Gastroenterology. 2023; 58(10):1131-1138. https://doi.org/10.1080/00365521.2023.2194010 IV. Diverticulosis and abdominal pain. Alexandersson B, Jones M, Hedin C, Schmidt PT, Andreasson A et al. Diverticulosis is not associated with more frequent abdominal pain, but it is more often located in the left lower quadrant – a population based Swedish cohort study. [Manuscript]</p
Schur polynomials, banded Toeplitz matrices and Widom's formula
No full textWe prove that for arbitrary partitions lambda subset of kappa, and integers 0 <= c < r <= n, the sequence of Schur polynomials S(kappa+k.1c)/(lambda+k.1r)(x(1), ... , x(n)) for k sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are also valid for certain sequences of minors of banded Toeplitz matrices. In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials.AuthorCount:1;</p
Mannen på gatan : Håkan Alexanderssons sista film
No full textShort analysis of the experimental short film "Mannen på gatan" ('The Man in the Street') by Swedish filmmaker Håkan Alexandersson (1940 - 2004)
New cases of the Strong Stanley Conjecture
No full textWe make progress towards understanding the structure of Littlewood-Richardson
coefficients for products of Jack symmetric functions.
Building on recent results of the second author, we are able to prove new cases
of a conjecture of Stanley in which certain families of these coefficients can
be expressed as a product of upper or lower hook lengths for every box in each
of the partitions. In particular, we prove that conjecture in the case of a
rectangular union, i.e. for where is the complementary partition of in the
rectangular partition . We give a formula for these coefficients through
an explicit prescription of such choices of hooks. Lastly, we conjecture an
analogue of this conjecture of Stanley holds in the case of Shifted Jack
functions.Comment: 23 pages, 26 figure
A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
No full textWe give a new characterization of the vertical-strip LLT polynomials GP(x;q) as the unique family of symmetric functions that satisfy certain combinatorial relations. This characterization is then used to prove an explicit combinatorial expansion of vertical-strip LLT polynomials in terms of elementary symmetric functions. Such formulas were conjectured independently by A. Garsia et al. and the first named author, and are governed by the combinatorics of orientations of unit-interval graphs. The obtained expansion is manifestly positive if q is replaced by q+1, thus recovering a recent result of M. D'Adderio. Our results are based on linear relations among LLT polynomials that arise in the work of D'Adderio, and of E. Carlsson and A. Mellit. To some extent these relations are given new bijective proofs using colorings of unit-interval graphs. As a bonus we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.</p
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