Soo-Kook Kim
Sungkyunkwan University · 数学
研究室紹介
Professor Soo-Kook Kim's research lab specializes in algebraic combinatorics, with a focus on symmetric functions, special polynomials, and their connections to polytopes, partitions, and orthogonal polynomials. The lab explores deep combinatorial structures such as noncrossing partitions, flow polytopes, and Young tableaux, often linking them to classical integrals like the Selberg integral and orthogonal polynomials such as Askey–Wilson. A central theme is the development of combinatorial proofs and interpretations for algebraic identities, including cyclic sieving phenomena and q-analogues of Catalan and Narayana numbers. The lab also investigates refined symmetric functions and their applications in representation theory and enumerative combinatorics.
Research Overview
Research Output Trend
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Selected Papers
15The Chan–Robbins–Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> . The normalized volume of the Chan–Ro
Abstract. The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects “Young books ” are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases of Young books become standard Young tableaux of various shapes: shifted staircases, square
We give combinatorial proofs of the formulas for the number of multichains in the $k-divisible$ noncrossing partitions of classical types with certain conditions on the rank and the block size due to Krattenthaler and Müller. We also prove Armstrong's conjecture on the zeta polynomial of the poset of k-divisible noncrossing partitions of type A invariant under the 180° rotation in the cyclic representation. Nous donnons une preuve combinatoire de la formule pour le nombre de multichaînes dans le
We show cyclic sieving phenomena on annular noncrossing permutations with given cycle types. We define annular q-Kreweras numbers, annular q-Narayana numbers, and annular q-Catalan numbers, and show that a sum of annular q-Kreweras numbers be- comes an annular q-Narayana number and a sum of annular q-Narayana numbers becomes an annular q-Catalan number. We also show that these polynomials are closely related to the cyclic sieving phenomena on annular noncrossing permutations.
Recently Galashin, Grinberg, and Liu introduced the refined dual stable Grothendieck polynomials, which are symmetric functions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>...</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with additional par
The mixed moments for the Askey–Wilson polynomials are found using a bootstrapping method and connection coefficients. A similar bootstrapping idea on generating functions gives a new Askey–Wilson generating function. Modified generating functions of orthogonal polynomials are shown to generate polynomials satisfying recurrences of known degree greater than three. An important special case of this hierarchy is a polynomial which satisfies a four term recurrence, and its combinatorics is studied.