Combinatorics of Macdonald Polynomials and Cyclic Sieving

70 Pages·2019·0.91 MB·English

by  Joakim Uhlin

DEGREE PROJECT IN MATHEMATICS,
SECOND CYCLE, 30 CREDITS
STOCKHOLM, SWEDEN 2019
Combinatorics of Macdonald
Polynomials and Cyclic
Sieving
JOAKIM UHLIN
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF ENGINEERING SCIENCES

Combinatorics of Macdonald
Polynomials and Cyclic Sieving
JOAKIM UHLIN
Degree Projects in Mathematics (30 ECTS credits)
Degree Programme in Mathematics (120 credits)
KTH Royal Institute of Technology year 2019
Supervisor at KTH: Per Alexandersson
Examiner at KTH: Svante Linusson
TRITA-SCI-GRU 2019:008
MAT-E 2019:03

Royal Institute of Technology
School of Engineering Sciences
KTH SCI
SE-100 44 Stockholm, Sweden
URL: www.kth.se/sci
2 JOAKIMUHLIN
Abstract
Inthisthesis,westudythenon-symmetricMacdonaldpolynomialsE (x;q,t)at
λ
t=0fromacombinatorialpointofview,usingthecombinatorialformulafoundby
J.Haglund, M.Haiman, andN.Loehr. Ourprimaryfocusiswhenλisapartition.
We summarize the known theory about this specialization and prove some new
results related to this combinatorial formula.
We also define the cyclic sieving phenomenon (CSP). For rectangular λ, we
presentaninstanceofcyclicsievingwithE (1,q,q2...,qk−1;1,0)asCSP-polynomial.
λ
WealsoconjectureanotherinstanceofCSPwithE (1,1,...,1;q,0)asCSP-polynomial.
λ
This conjecture generalizes a previously known CSP-triple. Furthermore, we prove
this conjecture in the case when λ is an m×2 diagram.

COMBINATORICS OF MACDONALD POLYNOMIALS AND CYCLIC SIEVING 3
Sammanfattning
Idennauppsatsstuderarvideicke-symmetriskaMacdonaldpolynomenE (x;q,t)
λ
n¨ar t=0 fr˚an en kombinatorisk synvinkel genom att anv¨anda den kombinatoriska
formeln som visats av J. Haglund, M. Haiman och N. Loehr. V˚art huvudsakliga
fokus ¨ar specialfallet d˚a λ ¨ar en partition. Vi summerar teorin om denna spe-
cialisering och visar ngra nya resultat som ¨ar relaterade till den kombinatoriska
formeln.
Vi definierar ¨aven det cykliska s˚allningsfenomenet (CSF). F¨or rektangul¨ara λ
visar vi en f¨orekomst av cyklisk s˚allning med E (1,q,q2,...,qk−1;1,0) som CSF-
λ
polynom. Vi presenterar ocks˚a en f¨ormodan om ytterligare en f¨orekomst av CSF
med E (1,1,...,1;q,0) som CSF-polynom. Denna f¨ormodan generaliserar en tidi-
λ
gare f¨orekomst av CSF. Vidare visar vi att denna f¨ormodan ¨ar sann i fallet d˚a λ ¨ar
ett m×2 diagram.

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Acknowledgement
Firstly, I cannot give a big enough thanks to my supervisor, Per Alexandersson.
His support, encouragement and seemingly endless supply of new ideas on how to
attack the problems I was facing made this thesis possible.
Secondly, I would like to thank all my friends in the masters program. With-
out you, studying would only be half as fun and my grades half as good.
Finally, I would like to thank Medile for her support (and patience) during the
work of this thesis.