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We investigate positivity and probabilistic properties arising from the Young-Fibonacci lattice $$\mathbb{YF}$$, a 1-differential poset on binary words composed of 1's and 2's (known as Fibonacci words). Building on Okada's theory of clone Schur functions (Trans. Amer. Math. Soc. 346 (1994), 549-568), we introduce clone coherent measures on $$\mathbb{YF}$$ which give rise to random Fibonacci words of increasing length. Unlike coherent systems associated to classical Schur functions on the Young lattice of integer partitions, clone coherent measures are generally not extremal on $$\mathbb{YF}$$. Our first main result is a complete characterization of Fibonacci positive specializations - parameter sequences which yield positive clone Schur functions on $$\mathbb{YF}$$. We connect Fibonacci positivity with total positivity of tridiagonal matrices, Stieltjes moment sequences, and orthogonal polynomials in one variable from the ($$q$$-)Askey scheme. Our second family of results concerns the asymptotic behavior of random Fibonacci words derived from various Fibonacci positive specializations. We analyze several limiting regimes for specific examples, revealing stick-breaking-like processes (connected to GEM distributions), dependent stick-breaking processes of a new type, or discrete limits tied to the Martin boundary of the Young-Fibonacci lattice. Our stick-breaking-like scaling limits significantly extend the result of Gnedin-Kerov (Math. Proc. Camb. Philos. Soc. 129 (2000), no. 3, 433-446) on asymptotics of the Plancherel measure on $$\mathbb{YF}$$. We also establish Cauchy-like identities for clone Schur functions (with the right-hand side given by a quadridiagonal determinant), and construct and analyze models of random permutations and involutions based on Fibonacci positive specializations and a version of the Robinson-Schensted correspondence for $$\mathbb{YF}$$.
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K-theoretic Catalan functions
We prove that the K-k-Schur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. Lam-Schilling-Shimozono identified the K-k-Schur functions as Schubert representatives for K-homology of the affine Grassmannian for SL_{k+1}. Our perspective reveals that the K-k-Schur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of Baldwin and Kumar. We further show that a slight adjustment of our formulation for K-k-Schur functions produces a second shift-invariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of Ikeda-Iwao-Maeno, we conjecture that this second basis gives the images of the Lenart-Maeno quantum Grothendieck polynomials under a K-theoretic analog of the Peterson isomorphism.
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- Award ID(s):
- 1840234
- PAR ID:
- 10530113
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Advances in Mathematics
- Volume:
- 404
- Issue:
- PB
- ISSN:
- 0001-8708
- Page Range / eLocation ID:
- 108421
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.aim.2022.108421
