Iterative character constructions for algebra groups

We construct a family of orthogonal characters of an algebra group which decompose the supercharacters defined by Diaconis and Isaacs. Like supercharacters, these characters are given by nonnegative integer linear combinations of Kirillov functions and are induced from linear supercharacters of certain algebra subgroups. We derive a formula for these characters and give a condition for their irreducibility; generalizing a theorem of Otto, we also show that each such character has the same number of Kirillov functions and irreducible characters as constituents. In proving these results, we observe as an application how a recent computation by Evseev implies that every irreducible character of the unitriangular group \UT_n(q) of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements is a Kirillov function if and only if $n\leq 12$. As a further application, we discuss some more general conditions showing that Kirillov functions are characters, and describe some results related to counting the irreducible constituents of supercharacters.Comment: 22 page

A materials informatics approach to the identification of one-band correlated materials analogous to the cuprates

One important yet exceedingly rare property of the cuprate high-temperature superconductors is the presence of a single correlated $d$ band in the low-energy spectrum, leading to the one-band Hubbard model as the minimal description. In order to search for materials with interesting strong correlation physics as well as possible benchmark systems for the one-band Hubbard model, here we present a new approach to find one-band correlated materials analogous to the cuprates by leveraging the emerging area of materials informatics. Using the composition, structure, and formation energy of more than half a million real and hypothetical inorganic crystalline materials in the Open Quantum Materials Database, we search for synthesizable materials whose nominal transition metal $d$ electron count and crystal field are compatible with achieving an isolated half-filled $d$ band. Five Cu compounds, including bromide, oxide, selenate, and pyrophosphate chemistries, are shown to successfully achieve the one-band electronic structure based on density functional theory band structure calculations. Further calculations including magnetism and explicit on-site Coulomb interaction reveal significant evidence for strong correlation physics in the five candidates, including Mott insulating behavior and antiferromagnetism. The success of our data-driven approach to discovering new correlated materials opens up new avenues to design and discover materials with rare electronic properties

Heisenberg characters, unitriangular groups, and Fibonacci numbers

Let \UT_n(\FF_q) denote the group of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements. We show that the Heisenberg characters of \UT_{n+1}(\FF_q) are indexed by lattice paths from the origin to the line $x+y=n$ using the steps $(1,0), (1,1), (0,1), (1,1)$, which are labeled in a certain way by nonzero elements of \FF_q. In particular, we prove for $n\geq 1$ that the number of Heisenberg characters of \UT_{n+1}(\FF_q) is a polynomial in $q-1$ with nonnegative integer coefficients and degree $n$, whose leading coefficient is the $n$th Fibonacci number. Similarly, we find that the number of Heisenberg supercharacters of \UT_n(\FF_q) is a polynomial in $q-1$ whose coefficients are Delannoy numbers and whose values give a $q$-analogue for the Pell numbers. By counting the fixed points of the action of a certain group of linear characters, we prove that the numbers of supercharacters, irreducible supercharacters, Heisenberg supercharacters, and Heisenberg characters of the subgroup of \UT_n(\FF_q) consisting of matrices whose superdiagonal entries sum to zero are likewise all polynomials in $q-1$ with nonnegative integer coefficients.Comment: 25 pages; v2: material significantly revised and condensed; v3: minor corrections, final versio