# Ricky Ini Liu

• Email: riliu (at) ncsu.edu
• Office: SAS Hall 3264
• Phone: 919-515-0681

### About me

I am currently an assistant professor at North Carolina State University. Previously I was a T. H. Hildebrandt Research Assistant Professor at the University of Michigan, Ann Arbor and an NSF postdoctoral fellow at the University of Minnesota, Twin Cities. I completed my Ph.D. at the Massachusetts Institute of Technology under the guidance of Alexander Postnikov in 2010. (CV)

My research interests lie in algebraic combinatorics, especially its relationship to algebraic geometry, combinatorial geometry, and representation theory.

### Papers and preprints

• Flow polytopes and the space of diagonal harmonics (with K. Mészáros and A. Morales), submitted.

We consider a generalization of Haglund's formula for the Hilbert series of the space of diagonal harmonics as a sum over Tesler matrices by summing over flow polytopes of threshold graphs.

• We show how the theory of noncommutative super Schur functions (based on work of Fomin and Greene) gives a combinatorial formula for the Kronecker coefficient $g_{\lambda\mu\nu}$ when $\mu$ is a hook.

• A simplified Kronecker rule for one hook shape, Proceedings of the American Mathematical Society, to appear.

We give a simple combinatorial rule for the Kronecker coefficient $g_{\lambda\mu\nu}$ when $\mu$ is a hook.

• Subalgebras of the Fomin-Kirillov algebra (with J. Blasiak and K. Mészáros), Journal of Algebraic Combinatorics, Volume 44, Issue 3, November 2016, pp. 785–829.

We define subalgebras of the Fomin-Kirillov algebra for any finite graph and show that they have a surprising number of nice properties analogous to those of Coxeter groups and nil-Coxeter algebras. We explicitly compute the Hilbert series for the subalgebras corresponding to the Dynkin diagrams $A_n$, $D_n$, $E_n$ ($n=6,7,8$), and $\tilde A_{n-1}$.

• On the commutative quotient of Fomin-Kirillov algebras, European Journal of Combinatorics, Volume 54, May 2016, pp. 65–75.

We prove that the commutative quotient of the Fomin-Kirillov algebra of a graph $G$ on $n$ vertices is isomorphic to the Orlik-Terao algebra of $G$. In particular, its Hilbert series is $(-t)^n \cdot \chi_G(-t^{-1})$, where $\chi_G(t)$ is the chromatic polynomial of $G$.

• Complete branching rules for Specht modules, Journal of Algebra, Volume 446, January 2016, pp. 77–102.

We give a combinatorial criterion for when the Specht module of an arbitrary diagram admits a (complete) branching rule. We also show that the only relations in such Specht modules are given by generalized Garnir relations.

• Positive expressions for skew divided difference operators, Journal of Algebraic Combinatorics, Volume 42, Issue 3, November 2015, pp. 861–874.

We give a positive formula for the skew divided difference operators (defined by Macdonald) in terms of divided difference operators $\partial_{ij}$ with $i < j$, settling a conjecture of Kirillov.

• Coefficients of a relative of cyclotomic polynomials, Acta Arithmetica, Volume 165, Number 4, 2014, pp. 301–325.

We give an explicit description of the coefficients of $(1-x)\Phi_{pqr}(x)$. We also define an analogous polynomial for any number of primes and describe their coefficients and growth rate as the number of primes increases.

• Laurent polynomials, Eulerian numbers, and Bernstein's theorem, Journal of Combinatorial Theory Series A, Volume 124, May 2014, pp. 244–250.

We give a simple proof of a result of Erman, Smith, and Várilly-Alvarado relating Laurent polynomials and Eulerian numbers using Bernstein's theorem. We also show that a refinement of the Eulerian numbers gives a combinatorial interpretation for volumes of certain (rational) hyperplane sections of the hypercube.

• Nonconvexity of the set of hypergraph degree sequences, Electronic Journal of Combinatorics, Volume 20, Issue 1, January 2013.

We show that the set of hypergraph degree sequences is not the intersection of a lattice and a convex polytope. We also prove an analogous result for multipartite hypergraphs.

• Matching polytopes and Specht modules, Transactions of the American Mathematical Society, Volume 364, Number 2, February 2012, pp. 1089–1107.

We prove that the normalized volume of the matching polytope of a forest equals the dimension of the corresponding Specht module. We also give $S_n$- and $GL_n$-branching rules, thereby defining a notion of Schur functions and standard/semistandard tableaux for forests.

• Matrices with restricted patterns and $q$-analogues of permutations (with J. Lewis, A. Morales, G. Panova, S. Sam, and Y. Zhang), Journal of Combinatorics, Volume 2, Number 3, 2011, pp. 355–395.

We study the number of matrices of a given rank with specified zero entries over a finite field. We show that these numbers give a $q$-analogue of certain restricted permutations and obtain explicit $q$-analogues for derangement numbers. We also consider the analogous questions for symmetric and skew-symmetric cases and discuss related questions.

• We present a conjecture relating the cohomology class of certain subvarieties of the Grassmannian to the structure of certain representations of the symmetric group and give evidence towards this conjecture.

• An algorithmic Littlewood-Richardson rule, Journal of Algebraic Combinatorics, Volume 31, Issue 2, February 2010, pp. 253–266.

We give a Littlewood-Richardson rule based on iteratively deforming a skew Young diagram into a straight shape. This rule is based on a geometric rule by Izzet Coskun.

• Counting subrings of $\mathbf Z^n$ of index $k$, Journal of Combinatorial Theory Series A, Volume 114, Issue 2, February 2007, pp. 278–299

We consider the problem of counting subrings of $\mathbf Z^n$ of a given index $k$. We show that a decomposition theorem holds and give a precise result when $n$ is at most 4 or $k$ is not divisible by the sixth power of any prime.

### Teaching

At North Carolina State University, I will be teaching the following course in Spring 2018:

Previously, I taught the following courses:

• Math 493: Combinatorial Game Theory, Fall 2017
• Math 416: Introduction to Combinatorics, Spring 2017
• Math 591: Algebraic Combinatorics, Fall 2016
• Math 724: Combinatorics II, Spring 2016
• Math 524: Combinatorics I, Fall 2015
• Math 416: Introduction to Combinatorics, Spring 2015
• Math 437: Applications of Algebra, Spring 2015

At the University of Michigan, I taught the following courses:

• Math 465: Introduction to Combinatorics, Winter 2014
• Math 215: Calculus III, Fall 2013
• Math 412: Introduction to Modern Algebra, Winter 2013
• Math 465: Introduction to Combinatorics, Fall 2012
• Math 217: Linear Algebra, Winter 2012
• Math 115: Calculus I, Fall 2011

In Fall 2009, I was a teaching assistant for 18.02: Multivariable Calculus at MIT.

I have been an instructor at the Mathematical Olympiad Summer Program since 2007. I was also a research adviser at Joe Gallian's Research Experience for Undergraduates at the University of Minnesota, Duluth for the summers 2006–2008.