THESIS
2022
1 online resource (ix, 78 pages) : illustrations (some color)
Abstract
In 1993, Fomin and Kirillov [4] introduced a new set of diagrams called RC graphs
or pipe dreams that aid in computing Schubert polynomials. Nowadays, there are
many studies on different kinds of pipe dreams and their bijections with other
combinatorial objects. Serrano and Stump [20] gave bijections between pipe
dreams, certain north-east fillings, and k-flagged tableaux. Hamaker, Marberg,
and Pawlowski [8] studied a new class of involution pipe dreams. Lam, Lee, and
Shimozono [13] more recently introduced bumpless pipe dreams and Gao and
Huang [6] have provided a bijection between bumpless pipe dreams and ordinary
pipe dreams.
As featured in the work of Knutson and Miller [12], there is a close connection
between pipe dreams and the geometry of Schubert varieties. Work of Marberg
and...[
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In 1993, Fomin and Kirillov [4] introduced a new set of diagrams called RC graphs
or pipe dreams that aid in computing Schubert polynomials. Nowadays, there are
many studies on different kinds of pipe dreams and their bijections with other
combinatorial objects. Serrano and Stump [20] gave bijections between pipe
dreams, certain north-east fillings, and k-flagged tableaux. Hamaker, Marberg,
and Pawlowski [8] studied a new class of involution pipe dreams. Lam, Lee, and
Shimozono [13] more recently introduced bumpless pipe dreams and Gao and
Huang [6] have provided a bijection between bumpless pipe dreams and ordinary
pipe dreams.
As featured in the work of Knutson and Miller [12], there is a close connection
between pipe dreams and the geometry of Schubert varieties. Work of Marberg
and Pawlowski [16] extends this connection to involution pipe dreams. Apart
from these geometric motivations, pipe dreams are also of interest on their own
in combinatorics.
This thesis first studies bijections between pipe dreams and a variety of other
objects: for example, certain north-east chains, north-east fillings, bumpless pipe dreams, and reverse plane partitions. This expository material surveys the main
results from [6] and [20]. Then we study symplectic variations of these constructions,
proving some new results. Specifically, we investigate bijections between
involution pipe dreams, certain reflected north-east chains and fillings, and
shifted reverse plane partitions. We also study a generalization of bumpless pipe
dreams for involutions. We give a conjectural bijection between these objects
and involution pipe dreams, extending the correspondence of [6].
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