#!/usr/bin/env python
# coding: utf-8

# # Counting indecomposable representations
# Given a quiver $Q$, we can compute its (absolutely) indecomposable representations over a finite field using the Hua formula. We use its formulation from [arXiv:math/0608321](https://arxiv.org/abs/math/0608321).
# 
# For any $d\in\Gamma^+=\mathbb N^{Q_0}$, we define its multi-partition to be a finite sequence of (nonzero) vectors $\lambda=(\lambda_1,\dots,\lambda_m)$ in $\Gamma^+$ such that $\lambda_1\ge\lambda_2\ge\dots\ge\lambda_m$ (meaning that $\lambda_k-\lambda_{k+1}\in\Gamma^+$) and $|\lambda|=\sum_k\lambda_k=d$.
# Define
# $$R_\lambda(q)=\prod_{k=1}^m\frac{q^{-\chi(\lambda_k,\lambda_k)}}{(q^{-1})_{\lambda_k-\lambda_{k+1}}},$$
# where $(q)_d=\prod_{i}(q)_{d_i}$ for $d\in\Gamma^+$ and $(q)_n=(1-q)\dots(1-q^n)$ for $n\in\mathbb N$. 
# Then the polynomials $A_d(q)$ counting absolutely indecomposable quiver representations having dimension vector $d$ are given by the formula
# $$\sum_d A_d(q)x^d=(q-1)Log\left(\sum_{\lambda}R_\lambda(q)x^{|\lambda|}\right),$$
# where $Log$ is the plethystic logarithm. We use polynomials in $q=y^2$.

# In[1]:


import os, sys
module_path = os.path.abspath(os.path.join('..'))
if module_path not in sys.path:
    sys.path.append(module_path)
from msinvar import *
from msinvar.indecomposable import hua_formula
set_plots()


# In[2]:


Q=JordanQuiver(1); show(Q)
Q.prec([3])
hua_formula(Q).dict()


# In[3]:


Q=Quiver('1-2'); show(Q)
Q.prec([3,3])
hua_formula(Q).dict()


# In[4]:


Q=KroneckerQuiver(2); show(Q)
Q.prec([3,3])
hua_formula(Q).dict()


# In[5]:


Q=KroneckerQuiver(2); show(Q)
Q.prec([3,3])
hua_formula(Q).dict()


# In[6]:


Q=CyclicQuiver(3); show(Q)
Q.prec([3,3,3])
hua_formula(Q).dict()


# In[ ]: