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Topics in Representation Theory
Topics in Representation Theory
Symplectic Skew Howe Duality, Christoffel Transformations, and Local Fluctuations in Random Young Diagrams
Symplectic Skew Howe Duality, Christoffel Transformations, and Local Fluctuations in Random Young Diagrams
Organizers
Speaker
Time
Friday, May 29, 2026 1:00 PM - 2:30 PM
Venue
A3-2a-302
Online
Zoom 242 742 6089
(BIMSA)
Abstract
We study local fluctuations of random Young diagrams with respect to probability measures arising from the skew $(Sp_{2n}, Sp_{2k})$ Howe duality. In the limit $n\to\infty$ with $k/n\to c$, the random Young diagrams converge to a deterministic limit shape. The diagrams are represented by random particle configurations described by an orthogonal polynomial ensemble whose correlation functions are determinantal and governed by the Christoffel–Darboux kernel for a family of semiclassical orthogonal polynomials obtained from the Krawtchouk polynomials by a doulbe Christoffel transformation.
To analyze the asymptotics of this kernel, we study Christoffel transformations for general orthogonal polynomials with symmetric weights. We prove that the Christoffel–Darboux kernel associated with the even-degree Christoffel-transformed polynomials is asymptotically close, in operator norm, to the even-degree kernel of the original polynomial family.
Applying this result to the skew $(Sp_{2n}, Sp_{2k})$ Howe duality ensemble, we identify four asymptotic regimes of local fluctuations. In addition to the universal bulk fluctuations governed by the discrete sine kernel and the universal Airy fluctuations at the right edge of the limit shape, we obtain the discrete Hermite kernel in the critical regime $k/n\to 1$ and the discrete symmetric sine kernel at the left edge.
To analyze the asymptotics of this kernel, we study Christoffel transformations for general orthogonal polynomials with symmetric weights. We prove that the Christoffel–Darboux kernel associated with the even-degree Christoffel-transformed polynomials is asymptotically close, in operator norm, to the even-degree kernel of the original polynomial family.
Applying this result to the skew $(Sp_{2n}, Sp_{2k})$ Howe duality ensemble, we identify four asymptotic regimes of local fluctuations. In addition to the universal bulk fluctuations governed by the discrete sine kernel and the universal Airy fluctuations at the right edge of the limit shape, we obtain the discrete Hermite kernel in the critical regime $k/n\to 1$ and the discrete symmetric sine kernel at the left edge.