" We further compute the dimension, stabilizer, and vertex set of standard parabolic faces of highest weight modules and show that they are completely determined by the aforementioned closed-form ...expressions. We also compute the f-polynomial and a minimal half-space representation of the convex hull of the set of weights. These results were recently shown for the adjoint representation of a simple Lie algebra, but analogues remain unknown for any other finite- or infinite-dimensional highest weight module. Our analysis is uniform and type-free, across all semisimple Lie algebras and for arbitrary highest weight modules.>
We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that ...are not the ``usual'' nil-Coxeter algebras: a novel 2-parameter type A family that we call NC_A(n,d). We explore several combinatorial properties of NC_A(n,d), including its Coxeter word basis, length function, and Hilbert-Poincaré series, and show that the corresponding generalized Coxeter group is not a flat deformation of NC_A(n,d). These algebras yield symmetric semigroup module categories that are necessarily not monoidal; we write down their Tannaka-Krein duality. Further motivated by the Broué-Malle-Rouquier (BMR) freeness conjecture J. Reine Angew. Math. 1998, we define generalized nil-Coxeter algebras NC_W over all discrete real or complex reflection groups W, finite or infinite. We provide a complete classification of all such algebras that are finite dimensional. Remarkably, these turn out to be either the usual nil-Coxeter algebras or the algebras NC_A(n,d). This proves as a special case--and strengthens--the lack of equidimensional nil-Coxeter analogues for finite complex reflection groups. In particular, generic Hecke algebras are not flat deformations of NC_W for W complex.
We correct an error in one of the main theorems (Theorem C) of “Standard parabolic subsets of highest weight modules”, Trans. Amer. Math. Soc. 369 (2017), no. 4.
Motivated by recent results of Tao–Ziegler
Discrete Anal.
2016 and Greenfeld–Tao (2022 preprint) on concatenating affine-linear functions along subgroups of an abelian group, we show three results ...on recovering affine linearity of functions
f
:
V
→
W
from their restrictions to affine lines, where
V
,
W
are
F
-vector spaces and
dim
V
⩾
2
. First, if
dim
V
<
|
F
|
and
f
:
V
→
F
is affine-linear when restricted to affine lines parallel to a basis and to certain “generic” lines through 0, then
f
is affine-linear on
V
. (This extends to all modules
M
over unital commutative rings
R
with large enough characteristic.) Second, we explain how a classical result attributed to von Staudt (1850 s) extends beyond bijections: If
f
:
V
→
W
preserves affine lines
ℓ
, and if
f
(
v
)
∉
f
(
ℓ
)
whenever
v
∉
ℓ
, then this also suffices to recover affine linearity on
V
, but up to a field automorphism. In particular, if
F
is a prime field
Z
/
p
Z
(
p
>
2
) or
Q
, or a completion
Q
p
or
R
, then
f
is affine-linear on
V
. We then quantitatively refine our first result above, via a weak multiplicative variant of the additive
B
h
-sets initially explored by Singer
Trans. Amer. Math. Soc.
1938, Erdös–Turán
J. London Math. Soc.
1941, and Bose–Chowla
Comment. Math. Helv.
1962. Weak multiplicative
B
h
-sets occur inside all rings with large enough characteristic, and in all infinite or large enough finite integral domains/fields. We show that if
R
is among any of these classes of rings, and
M
=
R
n
for some
n
⩾
3
, then one requires affine linearity on at least
n
⌈
n
/
2
⌉
-many generic lines to deduce the global affine linearity of
f
on
R
n
. Moreover, this bound is sharp.
The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on ...kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A. M. Whitney’s density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Pólya frequency functions, and Pólya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.
Entrywise functions preserving the cone of positive semidefinite matrices have been studied by many authors, most notably by Schoenberg Duke Math. J. 9, 1942 and Rudin Duke Math. J. 26, 1959. ...Following their work, it is well-known that entrywise functions preserving Loewner positivity in all dimensions are precisely the absolutely monotonic functions. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension 𝑛. Such characterizations for a fixed value of 𝑛 are difficult to obtain, and in fact are only known in the 2 × 2 case. In this paper, using a novel and intuitive approach, we study entrywise functions preserving positivity on distinguished submanifolds inside the cone obtained by imposing rank constraints. These rank constraints are prevalent in applications, and provide a natural way to relax the elusive original problem of preserving positivity in a fixed dimension. In our main result, we characterize entrywise functions mapping 𝑛 × 𝑛 positive semidefinite matrices of rank at most 𝑙 into positive semidefinite matrices of rank at most 𝑘 for 1 ≤ 𝑙 ≤ 𝑛 and 1 ≤ 𝑘 < 𝑛. We also demonstrate how an important necessary condition for preserving positivity by Horn and Loewner Trans. Amer. Math. Soc. 136, 1969 can be significantly generalized by adding rank constraints. Finally, our techniques allow us to obtain an elementary proof of the classical characterization of functions preserving positivity in all dimensions obtained by Schoenberg and Rudin.
Functions preserving Loewner positivity when applied entrywise to positive semidefinite matrices have been widely studied in the literature. Following the work of Schoenberg Duke Math. J. 9, Rudin ...Duke Math. J. 26, and others, it is well-known that functions preserving positivity for matrices of all dimensions are absolutely monotonic (i.e., analytic with nonnegative Taylor coefficients). In this paper, we study functions preserving positivity when applied entrywise to sparse matrices, with zeros encoded by a graph G or a family of graphs Gn
. Our results generalize Schoenberg and Rudin’s results to a modern setting, where functions are often applied entrywise to sparse matrices in order to improve their properties (e.g. better conditioning, graphical models). The only such result known in the literature is for the complete graph K
2. We provide the first such characterization result for a large family of noncomplete graphs. Specifically, we characterize functions preserving Loewner positivity on matrices with zeros according to a tree. These functions are multiplicatively midpoint-convex and superadditive. Leveraging the underlying sparsity in matrices thus admits the use of functions which are not necessarily analytic nor absolutely monotonic. We further show that analytic functions preserving positivity on matrices with zeros according to trees can contain arbitrarily long sequences of negative coefficients, thus obviating the need for absolute monotonicity in a very strong sense. This result leads to the question of exactly when absolute monotonicity is necessary when preserving positivity for an arbitrary class of graphs. We then provide a stronger condition in terms of the numerical range of all symmetric matrices, such that functions satisfying this condition on matrices with zeros according to any family of graphs with unbounded degrees are necessarily absolutely monotonic.
2010 Mathematics Subject Classification. Primary 15B48; Secondary 26E05, 05C50, 26A48.
Key words and phrases. Matrices with structure of zeros, entrywise positive maps, absolutely monotonic functions, multiplicatively convex functions, positive semidefiniteness, Loewner ordering, fractional Schur powers.
The Khinchin–Kahane inequality is a fundamental result in the probability literature, with the most general version to date holding in Banach spaces. Motivated by modern settings and applications, we ...generalize this inequality to arbitrary metric groups which are abelian. If instead of abelian one assumes the group's metric to be a norm (i.e., Z>0-homogeneous), then we explain how the inequality improves to the same one as in Banach spaces. This occurs via a “transfer principle” that helps carry over questions involving normed metric groups and abelian normed semigroups into the Banach space framework. This principle also extends the notion of the expectation to random variables with values in arbitrary abelian normed metric semigroups G. We provide additional applications, including studying weakly ℓpG-valued sequences and related Rademacher series. On a related note, we also formulate a “general” Lévy inequality, with two features: (i) It subsumes several known variants in the Banach space literature; and (ii) We show the inequality in the minimal framework required to state it: abelian metric groups.
PDF
