Combinatorial methods of character enumeration for the unitriangular group

Let \UT_n(q) denote the group of unipotent $n\times n$ upper triangular matrices over a field with $q$ elements. The degrees of the complex irreducible characters of \UT_n(q) are precisely the integers $q^e$ with $0\leq e\leq \lfloor \frac{n}{2} \rfloor \lfloor \frac{n-1}{2} \rfloor$, and it has been conjectured that the number of irreducible characters of \UT_n(q) with degree $q^e$ is a polynomial in $q-1$ with nonnegative integer coefficients (depending on $n$ and $e$). We confirm this conjecture when $e\leq 8$ and $n$ is arbitrary by a computer calculation. In particular, we describe an algorithm which allows us to derive explicit bivariate polynomials in $n$ and $q$ giving the number of irreducible characters of \UT_n(q) with degree $q^e$ when $n>2e$ and $e\leq 8$. When divided by $q^{n-e-2}$ and written in terms of the variables $n-2e-1$ and $q-1$, these functions are actually bivariate polynomials with nonnegative integer coefficients, suggesting an even stronger conjecture concerning such character counts. As an application of these calculations, we are able to show that all irreducible characters of \UT_n(q) with degree $\leq q^8$ are Kirillov functions. We also discuss some related results concerning the problem of counting the irreducible constituents of individual supercharacters of \UT_n(q).Comment: 34 pages, 5 table

Survey on counting special types of polynomials

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of multivariate polynomials over a finite field, namely the the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), the relatively irreducible ones (irreducible but reducible over an extension field), the decomposable ones, and also for reducible space curves. These come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size. Furthermore, a univariate polynomial f is decomposable if f = g o h for some nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic p of Fq does not divide n = deg f, is fairly well-understood, and we obtain closely matching upper and lower bounds on the number of decomposable polynomials. In the wild case, where p does divide n, the bounds are less satisfactory, in particular when p is the smallest prime divisor of n and divides n exactly twice. The crux of the matter is to count the number of collisions, where essentially different (g, h) yield the same f. We present a classification of all collisions at degree n = p^2 which yields an exact count of those decomposable polynomials.Comment: to appear in Jaime Gutierrez, Josef Schicho & Martin Weimann (editors), Computer Algebra and Polynomials, Lecture Notes in Computer Scienc

String Reconstruction from Substring Compositions

Motivated by mass-spectrometry protein sequencing, we consider a simply-stated problem of reconstructing a string from the multiset of its substring compositions. We show that all strings of length 7, one less than a prime, or one less than twice a prime, can be reconstructed uniquely up to reversal. For all other lengths we show that reconstruction is not always possible and provide sometimes-tight bounds on the largest number of strings with given substring compositions. The lower bounds are derived by combinatorial arguments and the upper bounds by algebraic considerations that precisely characterize the set of strings with the same substring compositions in terms of the factorization of bivariate polynomials. The problem can be viewed as a combinatorial simplification of the turnpike problem, and its solution may shed light on this long-standing problem as well. Using well known results on transience of multi-dimensional random walks, we also provide a reconstruction algorithm that reconstructs random strings over alphabets of size $\ge4$ in optimal near-quadratic time

Counting reducible, powerful, and relatively irreducible multivariate polynomials over finite fields

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible ones (irreducible but reducible over an extension field). One approach employs generating functions, another one uses a combinatorial method. They yield exact formulas and approximations with relative errors that essentially decrease exponentially in the input size.Comment: to appear in SIAM Journal on Discrete Mathematic

Symmetric nonnegative forms and sums of squares

We study symmetric nonnegative forms and their relationship with symmetric sums of squares. For a fixed number of variables $n$ and degree $2d$, symmetric nonnegative forms and symmetric sums of squares form closed, convex cones in the vector space of $n$-variate symmetric forms of degree $2d$. Using representation theory of the symmetric group we characterize both cones in a uniform way. Further, we investigate the asymptotic behavior when the degree $2d$ is fixed and the number of variables $n$ grows. Here, we show that, in sharp contrast to the general case, the difference between symmetric nonnegative forms and sums of squares does not grow arbitrarily large for any fixed degree $2d$. We consider the case of symmetric quartic forms in more detail and give a complete characterization of quartic symmetric sums of squares. Furthermore, we show that in degree $4$ the cones of nonnegative symmetric forms and symmetric sums of squares approach the same limit, thus these two cones asymptotically become closer as the number of variables grows. We conjecture that this is true in arbitrary degree $2d$.Comment: (v4) minor revision and small reorganizatio