[Paper Review] Relative symmetric polynomials and money change problem

M. Shahryari|arXiv (Cornell University)|Dec 14, 2010

Algebraic Geometry and Number Theory3 references5 citations

TL;DR

This paper establishes a novel connection between the money change problem—counting non-negative integer solutions to $ a_1t_1 + \cdots + a_nt_n = d $—and representation theory of the symmetric group $ S_m $, where $ m = \sum a_i $. Using relative symmetric polynomials and character theory, it derives an explicit formula for $ Q_d(a_1,\dots,a_n) $ as a sum over irreducible characters of $ S_m $, and provides a necessary and sufficient condition for the space of relative symmetric polynomials to be non-zero: $ H_d(S_m, \chi^\pi) \neq 0 $ iff there exists a composition $ \alpha \in \Gamma^{+}_{m,d} $ such that the multiplicity partition $ M(\alpha) $ majorizes $ \pi $. This links integer partition enumeration with symmetric group representation theory.

ABSTRACT

This article is devoted to the number of non-negative solutions of the linear Diophantine equation $$ a_1t_1+a_2t_2+... a_nt_n=d, $$ where $a_1, ..., a_n$, and $d$ are positive integers. We obtain a relation between the number of solutions of this equation and characters of the symmetric group, using {\em relative symmetric polynomials}. As an application, we give a necessary and sufficient condition for the space of the relative symmetric polynomials to be non-zero.

Motivation & Objective

• To establish a representation-theoretic framework for solving the classical money change problem using symmetric group characters.
• To connect the number of non-negative integer solutions to a linear Diophantine equation with the structure of relative symmetric polynomials.
• To derive a necessary and sufficient condition for the space of relative symmetric polynomials with respect to a character $ \chi^\pi $ of $ S_m $ to be non-zero.
• To express the solution count $ Q_d(a_1,\dots,a_n) $ as a character sum over irreducible representations of $ S_m $.

Proposed method

• Define $ Q_d(a_1,\dots,a_n) $ as the number of non-negative integer solutions to $ \sum a_i t_i = d $, with generating function $ \prod_{i=1}^n \frac{1}{1 - t^{a_i}} $.
• Model the solution space as a permutation character $ Q_d $ of the symmetric group $ S_m $, where $ m = \sum a_i $, acting on monomials of degree $ d $.
• Use the group algebra projector $ T(S_m, \chi^\pi) = \frac{\chi^\pi(1)}{m!} \sum_{\sigma \in S_m} \chi^\pi(\sigma) \sigma $ to define the space of relative symmetric polynomials $ H_d(S_m, \chi^\pi) $.
• Apply character inner product theory to express the dimension of $ H_d(S_m, \chi^\pi) $ as $ \chi^\pi(1) \cdot [\chi^\pi, Q_d]_{S_m} $.
• Use the induced character decomposition $ (1_{S_{M(\alpha)}})^{S_m} = \sum_{M(\alpha) \unlhd \pi} K_{\pi, M(\alpha)} \chi^\pi $, where $ K_{\pi, \mu} $ is the Kostka number.
• Derive the main formula: $ Q_d = \frac{1}{m!} \sum_{\alpha \in \Gamma^{+}_{m,d}} \sum_{M(\alpha) \unlhd \pi} M(\alpha)! \, K_{\pi, M(\alpha)} \, \chi^\pi $.

Experimental results

Research questions

• RQ1 How can the number of non-negative integer solutions to $ \sum a_i t_i = d $ be expressed in terms of symmetric group representation theory?
• RQ2 What is the relationship between the solution count $ Q_d $ and the irreducible characters of $ S_m $, where $ m = \sum a_i $?
• RQ3 Under what conditions is the space of relative symmetric polynomials $ H_d(S_m, \chi^\pi) $ non-zero?
• RQ4 Can the generating function for $ Q_d $ be interpreted as a character of $ S_m $, and if so, how is it decomposed into irreducibles?

Key findings

• The solution count $ Q_d(a_1,\dots,a_n) $ is equal to the character of the permutation representation of $ S_m $ acting on monomials of degree $ d $, and thus is a permutation character.
• The dimension of the space of relative symmetric polynomials $ H_d(S_m, \chi^\pi) $ is given by $ \chi^\pi(1) \cdot [\chi^\pi, Q_d]_{S_m} $, which equals $ \frac{\chi^\pi(1)}{m!} \sum_{\alpha \in \Gamma^{+}_{m,d}, M(\alpha) \unlhd \pi} M(\alpha)! \, K_{\pi, M(\alpha)} $.
• The function $ Q_d $ decomposes as $ Q_d = \frac{1}{m!} \sum_{\alpha \in \Gamma^{+}_{m,d}} \sum_{M(\alpha) \unlhd \pi} M(\alpha)! \, K_{\pi, M(\alpha)} \, \chi^\pi $, expressing the solution count as a sum over irreducible characters of $ S_m $.
• The space $ H_d(S_m, \chi^\pi) $ is non-zero if and only if there exists a composition $ \alpha \in \Gamma^{+}_{m,d} $ such that the multiplicity partition $ M(\alpha) $ majorizes $ \pi $.
• The Kostka number $ K_{\pi, M(\alpha)} $ is non-zero precisely when $ M(\alpha) \unlhd \pi $, which is the key condition for the irreducible character $ \chi^\pi $ to appear in the decomposition of $ Q_d $.