Upper bounds of dual flagged Weyl characters

Schubert polynomials are refined by the key polynomials of Lascoux-Schützen-berger, which in turn are refined by the fundamental slide polynomials of Assaf-Searles. In this paper we determine which fundamental slide polynomial refinements of key polynomials, indexed by strong compositions, are multiplicity free. We also give a recursive algorithm to determine all terms in the fundamental slide polynomial refinement of a key polynomial indexed by a strong composition. From here, we apply our results to begin to classify which fundamental slide polynomial refinements, indexed by weak compositions, are multiplicity free. We completely resolve the cases when the weak composition has at most two nonzero parts or the sum has at most two nonzero terms.

Vertices of Schubitopes

Let w be a permutation of {1, 2, …, n}, and let D(w) be the Rothe diagram of w. The Schubert polynomial \({\mathfrak{S}_w}\left(x \right)\) can be realized as the dual character of the flagged Weyl module associated with D(w). This implies the following coefficient-wise inequality: $${\rm{Mi}}{{\rm{n}}_w}\left(x \right) \le {\mathfrak{S}_w}\left(x \right) \le {\rm{Ma}}{{\rm{x}}_w}\left(x \right),$$where both Minw(x) and Maxw(x) are polynomials determined by D(w). Fink et al. (2018) found that \({\mathfrak{S}_w}\left(x \right)\) equals the lower bound Minw(x) if and only if w avoids twelve permutation patterns. In this paper, we show that \({\mathfrak{S}_w}\left(x \right)\) reaches the upper bound Maxw(x) if and only if w avoids two permutation patterns 1432 and 1423. Similarly, for any given composition α ∈ ℤ n≽0 , one can define a lower bound Minα(x) and an upper bound Maxα(x) for the key polynomial κα(x). Hodges and Yong (2020) established that κα(x) equals Minα(x) if and only if α avoids five composition patterns. We show that κα(x) equals Maxα(x) if and only if α avoids a single composition pattern (0, 2). As an application, we obtain that when α avoids (0, 2), the key polynomial κα(x) is Lorentzian, partially verifying a conjecture of Huh et al. (2019).

We prove a criterion of when the dual character $\chi _{D}(x)$ of the flagged Weyl module associated a diagram D in the grid $[n]\times [n]$ is zero-one, that is, the coefficients of monomials in $\chi _{D}(x)$ are either 0 or 1. This settles a conjecture proposed by Mészáros–St. Dizier–Tanjaya. Since Schubert polynomials and key polynomials occur as special cases of dual flagged Weyl characters, our approach provides a new and unified proof of known criteria for zero-one Schubert/key polynomials due to Fink–Mészáros–St. Dizier and Hodges–Yong, respectively.

Flagged Schur functions, Schubert polynomials, and symmetrizing operators

Schubert polynomials as integer point transforms of generalized permutahedra

Local Weyl modules over two-dimensional currents with values in Glr are deformed into spaces with bases related to parking functions. Using this construction, we (1) propose a simple proof that dimension of the space of diagonal coinvariants is not less than the number of parking functions; (2) describe the limits of Weyl modules in terms of semi-infinite forms and find the limits of characters; and (3) propose a lower bound and state a conjecture for dimensions of Weyl modules with arbitrary highest weights. Also we express characters of deformed Weyl modules in terms of ρ-parking functions and the Frobenius characteristic map.

Schur polynomials are special cases of Schubert polynomials, which in turn are special cases of dual characters of flagged Weyl modules. The principal specialization of Schur and Schubert polynomials has a long history, with Macdonald famously expressing the principal specialization of any Schubert polynomial in terms of reduced words. We prove a lower bound on the principal specialization of dual characters of flagged Weyl modules. Our result yields an alternative proof of a conjecture of Stanley about the principal specialization of Schubert polynomials, originally proved by Weigandt.

Tropicalization, symmetric polynomials, and complexity

For a Chevalley group G over an algebraically closed field K of characteristic p>0 with the irreducible root system R, Lusztig’s character formula expresses the formal character of a simple G-module by the formal characters of the Weyl modules and the values of the Kazhdan–Lusztig polynomials at 1. It is known that, for a sufficiently large characteristic p of the field K, Lusztig’s character formula holds. The known lower bound of the characteristic p is much larger than the Coxeter number h of the root system R. Observations show that for simple modules with restricted highest weights of small Chevalley groups such as those of types A1,A2, A3,B2, B3, and C3, Lusztig’s character formula holds for all p≥h. For large Chevalley groups, no other examples are known. In this paper, for G of type Al, we give some series of simple modules for which Lusztig’s character formula holds for all p≥h. Using this result, we compute the cohomology of G with coefficients in these simple modules. To prove the results, Jantzen’s filtration properties for Weyl modules and the properties of Kazhdan–Lusztig polynomials are used.

In a previous paper (1988), the author proposed methods to obtain sharp lower and upper bounds for probabilities that at least one out of n events occurs, based on the knowledge of some of the binomial moments of the number of events which occur and linear programming formulations. This paper presents further results concerning other and more general logical functions of events: We give sharp lower and upper bounds for the probabilities that: a) exactly r events, b) at least r events occur, using linear programming. The basic facts are expressed by the dual feasible basis characterization theorems which are interpreted in terms of the vertices of the dual problems. We mention some linear inequalities, among the binomial moments, generalize the theory for the case of nonconsecutive binomial moments and present numerical examples.

A Lagrangean heuristic for the capacitated concave minimum cost network flow problem

This paper studies the mixed structured singular value, /spl mu/, and the well-known (D,G)-scaling upper bound, /spl nu/. A dual characterization of /spl mu/ and /spl nu/ is derived, which intimately links the two values. Using the duals it is shown that /spl nu/ is guaranteed to be lossless (i.e. equal to /spl mu/) if and only if 2(m/sub r/+m/sub e/)+m/sub C//spl les/3, where m/sub r/, m/sub c/; and m/sub C/ are the numbers of repeated real scalar blocks, repeated complex scalar blocks, and full complex blocks, respectively. The losslessness result further leads to a variation of the well-known Kalman-Yakubovich-Popov lemma and Lyapunov inequalities.

This paper studies the mixed structured singular value, /spl mu/, and the well-known (D,G)-scaling upper bound, /spl nu/. A complete characterization of the losslessness of /spl nu/ (i.e., /spl nu/ being equal to /spl mu/) is derived in terms of the numbers of different perturbation blocks. Specifically, it is shown that /spl nu/ is guaranteed to be lossless if and only if 2(m/sub r/+m/sub c/)+m/sub C//spl les/3, where m/sub r/, m/sub c/ and m/sub C/ are the numbers of repeated real scalar blocks, repeated complex scalar blocks and full complex blocks, respectively. The results hinge on a dual characterization of /spl mu/ and /spl nu/ which intimately links /spl mu/ with /spl nu/. Further, a special case of the aforementioned losslessness result leads to a variation of the well-known Kalman-Yakubovich-Popov lemma and Lyapunov inequalities.