Arithmetic of Heisenberg ring and cyclic group actions

TL;DR

This paper investigates the structure of the Heisenberg ring's configuration space by analyzing fiber cardinalities through cyclic group actions on restricted partitions, providing new computational insights.

Contribution

It introduces novel methods for computing fiber sizes in Heisenberg ring configuration spaces using cyclic group actions on partitions.

Findings

Computed fiber cardinalities in new cases

Established connections between group actions and partition structures

Enhanced understanding of Heisenberg ring configuration space

Abstract

In this paper we compute in some new cases the cardinalities of the fibers of certain natural fibrations that appear in the analysis of the configuration space of the Heisenberg ring. This is done by means of certain cyclic group actions on some subsets of restricted partitions.

Equations

$$\tilde{N}=\{j|j=1,\dots ,N\}\equiv \mathbb{Z}/N\mathbb{Z},\mathrm{and},[j{]}_N\in \mathbb{Z}/N\mathbb{Z},1\le j\le N.$$

$$Q_{N,r}=\{\mathbf{j}=(j_1,\dots ,j_r)\in {\tilde{N}}^r|1\le j_1<\dots <j_r\le N\}$$

$$P_{N,r}=\{(j_1,\dots ,j_r)\in {\mathbb{Z}}^r|j_1<\dots <j_r<j_1+N\}$$

$$L(1,(j_1,\dots ,j_r))=(j_2,\dots ,j_r,j_1+N).$$

$$Q_{N,r}\equiv P_{N,r}/L.$$

$${\pi }_{N,r}:Q_{N,r}⟶\mathbb{Z}/N\mathbb{Z},{\pi }_{N,r}(\mathbf{j})={\left[\sum _{\alpha \in \tilde{r}}{j}_{\alpha }\right]}_N,$$

$${\tilde{\pi }}_{N,r}:P_{N,r}⟶\mathbb{Z},{\tilde{\pi }}_{N,r}(\mathbf{j})=\sum _{\alpha \in \tilde{r}}{j}_{\alpha }$$

$${\Delta }_{N,r}=\{\mathbf{t}\in {\mathbb{N}}^r|\sum _{\alpha \in \tilde{r}}{t}_{\alpha }=N\}$$

$$t_{\alpha }=\{\begin{array}{l}{j}_{\alpha +1}-j_{\alpha }\mathrm{for}1\le \alpha <r,\\ N+j_1-j_r\mathrm{for}\alpha =r\end{array},$$

$${\Delta }_{N,r}=\{\mathbf{t}\in {\mathbb{N}}^r|\sum _{\alpha \in \tilde{r}}{t}_{\alpha }=N\}.$$

$$j_{\alpha }(s,\mathbf{t})=\frac{1}{r}(s+\sum _{\beta \in \tilde{r}}\beta t_{\beta })-\sum _{\alpha \le \beta \le r}{t}_{\beta },\alpha \in \tilde{r}.$$

$$s+\sum _{\beta \in \tilde{r}}\beta t_{\beta }\equiv 0\mathrm{mod}r.$$

$$Q_{N,r}=\bigcup _{s\in \mathbf{Z}/N\mathbf{Z}}{Q}_{N,r,s},Q_{N,r,s}:=\{\mathbf{j}\in Q_{N,r}|{\pi }_{N,r}(\mathbf{j})=s\}$$

$$P_{N,r}=\bigcup _{s\in \mathbf{Z}}{P}_{N,r,s},P_{N,r,s}:=\{\mathbf{j}\in P_{N,r}|{\tilde{\pi }}_{N,r}(\mathbf{j})=s\}.$$

$${\Delta }_{N,r}=\bigcup _{s\in \mathbf{Z}/N\mathbf{Z}}{\Delta }_{N,r,s},{\Delta }_{N,r,s}:=\{\mathbf{t}\in {\Delta }_{N,r}|s+\sum _{\beta \in \tilde{r}}\beta t_{\beta }\equiv 0\mathrm{mod}r\}.$$

$${\Delta }_{Mp,p,0}={\Delta }_{Mp,p,s}+1.$$

$$\mathfrak{p}(M,N,n)=\mathfrak{p}(M,N-1,n)+\mathfrak{p}(M-1,N,n-N)$$

$$G(L,K;q)={\left[\genfrac{}{}{0ex}{}{K+L}{K}\right]}_q=\frac{{\prod }_{j=1}^{K+L}(1-q^j)}{{\prod }_{j=1}^K(1-q^j){\prod }_{j=1}^L(1-q^j)}$$

$${\sum }_{n=0}^{MN}\mathfrak{p}(M,N,n)q^n={\left[\genfrac{}{}{0ex}{}{M+N}{N}\right]}_q$$

$${\left[\genfrac{}{}{0ex}{}{N}{M}\right]}_q={\left[\genfrac{}{}{0ex}{}{N-1}{M-1}\right]}_q+q^M{\left[\genfrac{}{}{0ex}{}{N-1}{M}\right]}_q$$

$$\begin{array}{rl}{}^r{T}_{k,l}^i:=& \{\text{the set of restricted partitions of a number congruent to}i\mathrm{mod}r\\ & \text{that fit into the rectangle k×l }\}\end{array}$$

$$T_{k,l}={}^r{T}_{k,l}^0\bigsqcup {}^r{T}_{k,l}^1\bigsqcup \cdots \bigsqcup {}^r{T}_{k,l}^{r-1}$$

$${}^r{\Omega }_{k,l}^i=\{\text{non-increasing surjections}f:[0,k]\to \{1,\dots ,l\}:{\int }_0^{k}fdx\equiv i\mathrm{mod}r\}.$$

$$f:{}^r{\Omega }_{k+l,l}^m\to {}^r{T}_{k,l-1}^i$$

$$f((t_l,\dots ,t_2,t_1))=<t_2-(l-1),\dots ,(t_l-1)>=<{\pi }_1,\dots {\pi }_{l-1}>$$

$$|T_{k,l}|=\left(\genfrac{}{}{0ex}{}{k+l}{l}\right)$$

$$(n_1,n_2,\dots ,n_l)\text{where}{n}_1=t_l,n_2=t_{l-1}-t_l,\dots ,n_l=k+l-t_2$$

$$(t_2,\dots ,t_l)\text{such that}0<t_l<t_{l-1}<\cdots <t_2<k+l$$

$$(n_1,\dots ,n_l)\text{such that}{n}_i>0\text{and}{n}_1+n_2+\dots n_l=k+l.$$

$$C\cdot (n_1,n_2,\dots ,n_l)=(n_l,n_1,\dots ,n_{l-1})$$

Full text

†† 2000 Mathematics Subject Classification. 05 A17, 05A19, 05E18. †† Key words and phrases. Heisenberg ring, partitions of integers, Gaussian polynomials, group actions

1 Introduction

In this paper we consider the Heisenberg ring with $N$ nodes and $r\leq\frac{N}{2}$ reversed spins. According to this model at each node at most one reverse is admissible. Any such reverse is called a Bethe pseudo-particle. By assumption Bethe pseudo-partcles are identical and indistinguishable. It is natural to treat the set of nodes as a cyclic group ${\mathbb{Z}}/n{\mathbb{Z}}$ or rather a regular orbit of it. Thus

[TABLE]

Classical configuration space of the Heisenberg ring with $r$ Bethe pseudo-particles can be identified with the following set:

[TABLE]

which corresponds, under the identification (1.1), to the family of $r$ element subsets of $\Z/n\Z$.

On the covering space:

[TABLE]

we introduce an action $L\,:\,({\Z},+)\times P_{N,r}\longrightarrow P_{N,r}$ defined by the following condition:

[TABLE]

The orbit space of this action is equal to the classical configuration space:

[TABLE]

The following projections:

[TABLE]

[TABLE]

can be interpreted as suitably rescaled centers of masses of configurations. More precisely, they are multiplied by the number of Bethe pseudo-particles. The set of all relative positions

[TABLE]

where ${\bf t}={\bf t}({\bf j})=(t_{1},\ldots,t_{r})$ is the vector of relative positions with the following components:

[TABLE]

forms a lattice simplex:

[TABLE]

Each configuration ${\bf j}\in P_{N,r}$ can be uniquely determined by means of the aforementioned rescaled center of mass $s=\pi_{N,r}({\bf j})$ and the relative configuration ${\bf t}({\bf j})$:

[TABLE]

From this we obtain the compatibility condition for $s$ and ${\bf t:}$

[TABLE]

The projections $\pi_{N,r}$ resp. $\tilde{\pi}_{N,r}$ are fibrations with the total spaces spaces $Q_{N,r}$ resp. $P_{N,r}$ . We have

[TABLE]

and

[TABLE]

In [5] it was shown that for any $N\geq 2$ the configuration space $Q_{N,r}$ can be naturally embedded, in a natural way, in a model manifold $Q_{r}(S)=\{A\subset S\,|\,\,cardA=r\}.$ An analogous to considered above coverings and fibrations were also given. These constructions resemble the fibrations considered by H. R. Morton [7] and D.Tymoczko [10] (cf. also [11]) who applied them in the theory of music chords. They analyze configurations with repetitions which lead to orbifolds. Since we consider configurations without repetitions, our approach yields manifolds.

With these fibrations one can associate one more fibration (cf. the compatibility condition (1.12)):

[TABLE]

In the paper [6] the number of elements in these fibers for some values of $N$ were computed. It was shown, using certain natural group action, that for $N,r$ coprime the fibers have the same number of elements. Similarly, for $s\neq 0\mod r$ all fibers $\Delta_{Mp,p,s}$ are of equal cardinality for a rational prime $p>2$ and $M>1$. For $p=3,5$ and $s\neq 0\mod r$, it was additionally shown that

[TABLE]

In this paper we show that the above formula holds for any prime $p>2.$

We phrase the relations in terms of the relation among coefficients of Gaussian polynomials (cf. Remark 3.2 and Lemma 3.3). The cardinalities of the fibers $|\Delta_{Mp,p,s}|,\,s=0,\dots,p-1$ follow then directly from Theorem 4.13

For known relations for coefficients of Gaussian polynomials and history see [4]. In this short note we give a direct combinatorial proof of some relations between the sums of coefficients of Gaussian polynomials. We prove that if $\gcd(k,l)=1$ then the sums over moduli classes modulo $l$ of coefficients of Gaussian polynomials corresponding to the rectangle $k\times(l-1)$ are all equal (cf. Theorem 4.3). We also prove one case where $\gcd(k,l)\neq 1.$ Namely for an odd prime number $p$ and $k=Mp,l=p$ we compute the corresponding sums. This result is not true when $p$ is not a prime. The main tool is the action of the additive group ${\mathbb{Z}}/p{\mathbb{Z}}$ on certain subsets of the restricted partitions. In section 2 we very briefly give necessary definitions. In section 3 we define some group actions on the sets of restricted partitions. We also give some comments and state a general problem.

2 Preliminaries

In this section we set up notation and recall basic facts concerning partitions. The reader is advised to consult one of the excellent references on the subject [1], [2],[8], [9]. Let ${\mathfrak{p}}(N,M;n)$ denote the number of partitions of a natural number $n$ into at most $M$ parts, each of size not exceeding $N$. Geometrically that means that the Young diagram corresponding to such a partition $\pi=<{\pi}_{1},\dots,{\pi}_{l}>\,,\quad\,{\pi}_{1}\geq,\dots,\geq{\pi}_{l}$ fits inside an $M\times N$ rectangle. The following recurrence formula holds

[TABLE]

Formula (2.17) comes from the observation that there is a bijection between the partitions of $n$ into exactly $N$ parts of size not exceeding $M$ and the partitions of $n-N$ into at most N parts of size not exceeding $M-1.$ The bijection is given by subtracting $1$ from each part of the former.

The Gaussian coefficient may be considered as a generalization of the usual binomial coefficient and is defined by the following formula:

[TABLE]

The Gaussian binomial coefficient yields a generating function of restricted partitions

[TABLE]

In particular (2.19) shows that the Gaussian binomial coefficient is a polynomial with integer coefficients. Equation (2.17) yields the following

[TABLE]

3 Some group actions

Let $k,l,r$ be natural numbers and $i\in\{0,1,\dots,r-1\}.$ Define the following sets

[TABLE]

Let

[TABLE]

be the set of partitions that fit into the rectangle $k\times l.$ Notice that $T_{k,l}$ is independent of $r$ and of course $|T_{k,l}|=|T_{l,k}|.$ Let further

[TABLE]

and ${\Omega}_{k,l}={{}^{r}}{\Omega}^{0}_{k,l}\sqcup\dots\sqcup{{}^{r}}{\Omega}^{r-1}_{k,l}.$ In (3.23) $[0,k]\subset{\mathbb{R}}$ denotes the closed interval.

Remark 3.1**.**

Notice that ${\Omega}_{k,l}$ is the set of all non-increasing surjections $f:[0,k]\rightarrow\{1,\dots,l\}$ and thus is independent of $r.$

Remark 3.2**.**

Notice that by (1.15) we have a bijection between ${\Delta}_{N,r,s}$ and ${{}^{r}}{\Omega}^{r-s}_{N,r}$ and therefore $|{\Delta}_{N,r,s}|=|{{}^{r}}{\Omega}^{r-s}_{N,r}|.$

We start with the following lemma.

Lemma 3.3**.**

Let ${m=l(l-1)/2}+k+l+i$ There is a bijection between the following sets

[TABLE]

Proof..

Any non-increasing surjection in ${\Omega}_{k+l,l}$ is uniquely defined by a sequence of natural numbers $0<t_{l}<t_{l-1}<\dots<t_{2}<t_{1}=k+l.$ Order these sequences lexicographically. Among them there is a minimal sequence (in the left diagram of Fig.1 the minimal surjection corresponds to the grey region, for k= 11, l=4 ). There is also a maximal element which corresponds to the border of the crossed region. Any non-increasing surjection has to have its graph inside the region between the grey and crossed regions. In Fig.1 we depicted the surjection corresponding to the sequence $(t_{4},t_{3},t_{2},t_{1})=(4,5,7,15).$ We define the map $f$ in the following way

[TABLE]

where the sequence in the brackets denotes the partition assigned to a non-increasing surjection (cf. Fig.1). This gives the bijection (3.24) and also a bijection between ${\Omega}_{k+l,l}$ and $T_{k,l-1}.$

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*(lightgray) ∙ *(lightgray) ∙∙∙∙ × × \none∙∙ ∙

*(lightgray) ∙ *(lightgray) ∙ *(lightgray) ∙∙∙∙∙ × \none∙ ∙ ∙∙

(lightgray) ∙ (lightgray) ∙ (lightgray) ∙(lightgray) ∙(lightgray) ∙(lightgray) ∙(lightgray) ∙(lightgray) ∙ *(lightgray) ∙ *(lightgray) ∙ *(lightgray) ∙ (lightgray) ∙(lightgray) ∙ (lightgray) ∙(lightgray) ∙\none\none\none\none \none \none \none \none\none \none\none

Fig.1

∎

Notice that the partition $\bf 0\in\rm{{}^{r}}T_{k,l}^{0}$, for any $r$, and corresponds to the minimal surjection.

Corollary 3.4**.**

The number of elements in the set $T_{k,l}$

[TABLE]

Proof..

By Lemma 3.3 $|T_{k,l}|=|{\Omega}_{k+l+1,l+1}|$ and elements of ${\Omega}_{k+l+1,l+1}$ correspond bijectively to the sequences $0<t_{l+1}<t_{l}<\dots<t_{2}<k+l+1$ ∎

Let a sequence $0<t_{l}<t_{l-1}<\dots<t_{2}<k+l$ correspond to an element $f\in{\Omega}_{k+l,l}$ as in the proof of Corollary 3.4 .

Let

[TABLE]

be the corresponding sequence of lengths of subintervals of $[0,k+l].$ Notice that assignment (3.27) gives a bijection between the following sets of sequences

[TABLE]

and

[TABLE]

We have the following group actions on the sets ${\Omega}_{k+l,l}$. We define them on the sequences (3.27).

Action of the cyclic group ${\mathbb{Z}}/{l{\mathbb{Z}}}$

Let $C$ be a generator of the cyclic group ${\mathbb{Z}}/{l{\mathbb{Z}}}.$ We define the action of ${\mathbb{Z}}/{l{\mathbb{Z}}}$ on the sequence $(n_{1},n_{2},\dots,n_{l})$ by the following formula

[TABLE]

Action of the group of units $({\mathbb{Z}}/{l{\mathbb{Z}}})^{*}$

Let $u\in({\mathbb{Z}}/{l{\mathbb{Z}}})^{*}.$ Define the action of $u$ on the sequence $(n_{1},\dots n_{l})$ by the following formula

[TABLE]

Action of the symmetric group ${\cal S}_{l}$

Let ${\cal S}_{l}$ be the symmetric group on $l$ letters we define the action of ${\sigma}\in{\cal S}_{l}$ on a sequence $(n_{1},n_{2},\dots,n_{l})$ by the formula

[TABLE]

These actions, via bijection $g:{\Omega}_{k+l,l}\rightarrow T_{k,l-1}$ of Lemma 3.3, yield the corresponding action on the set of restricted partitions $T_{k,l-1}$. This transfer of actions is given by the following formula

[TABLE]

where $\alpha$ is an element of an appropriate group.

These actions on the sets of restricted partitions are of their own interest. We shall use the action (3.30) for obtaining some equalities concerning coefficients of Gaussian polynomials. We shall also use a different action of the additive group ${\mathbb{Z}}/p{\mathbb{Z}}$ on certain subsets of restricted partitions to obtain similar equalities for odd prime numbers.

4 Main results

We have the following lemma

Lemma 4.1**.**

Let $k,l>0$ be natural numbers. If $\gcd(k,l)=1$ then for any $i,j\in{\mathbb{Z}}/l{\mathbb{Z}}$ we have $|{{}^{l}}{\Omega}_{k+l,l}^{i}|=|{{}^{l}}{\Omega}_{k+l,l}^{j}|.$

Proof..

We use the action (3.30). Notice that for $f\in{{}^{l}}{\Omega}_{k+l,l}$ we have ${\int}_{0}^{k+l}fdx=n_{1}\cdot l+n_{2}\cdot(l-1)+\dots+n_{l}.$ Thus if $n_{1}\cdot l+n_{2}\cdot(l-1)+\dots+n_{l}\equiv\,i\,\mod\,l$ then

[TABLE]

[TABLE]

Since $\gcd(k+l,l)=1,$ we see that the set of congruence classes of integrals of the functions in the orbit $\{f,Cf,\dots C^{l-1}f\}$ correspond bijectively to ${\mathbb{Z}}/l{\mathbb{Z}}.$ This proves the lemma. ∎

Notice that Lemma 4.1, although formulated in different context, is similar to Lemma 1 of [6].

Corollary 4.2**.**

Let $k,l>0$ be natural numbers such that $\gcd(k,l)=1$ and let $r$ be a divisor of $l.$ Then $|{{}^{r}}{\Omega}_{k,l}^{i}|=|{{}^{r}}{\Omega}_{k,l}^{j}|$ for any $i,j\in{\mathbb{Z}}/r{\mathbb{Z}}.$

Proof..

Apply Lemma 4.1 and the reduction map ${\mathbb{Z}}/l{\mathbb{Z}}\rightarrow{\mathbb{Z}}/r{\mathbb{Z}}.$ ∎

Theorem 4.3**.**

Let $k,l$ be positive integers such that $\gcd(k,l)=1$ and $r$ any positive divisor of $l.$ For any $i\in{\mathbb{Z}}/r{\mathbb{Z}}$ we have

[TABLE]

Proof..

Corollary 4.2 and Lemma 3.3 show that under the assumption of the theorem for any $i,j\in{\mathbb{Z}}/r{\mathbb{Z}}$ we have $|{{}^{r}}T_{k,l-1}^{i}|=|{{}^{r}}T_{k,l-1}^{j}|.$ The left hand side of (4.35) is equal to $|{{}^{r}}T_{k,l-1}^{i}|.$ Now the theorem follows from Corollary 3.4. ∎

Example 4.4**.**

For $k=3,l=4$ and $r=4$ we have $T_{k,l-1}^{i}=5$ for $i=0,1,2,3.$

Example 4.5**.**

For $k=5,l=3$ and $r=3$ we have $T_{k,l-1}^{i}=7$ for $i=0,1,2.$

Remark 4.6**.**

Notice that the left hand side of (4.35) is the sum ${\sum}_{{n\equiv r}({\text{mod}}p)}a_{n}$ over the moduli classes modulo $r$ of the coefficients of the Gaussian polynomial $G(k,l-1,q)={\sum}_{n=0}^{k(l-1)}a_{n}q^{n}.$

Remark 4.7**.**

Theorem 4.3 is false if $\gcd(k,l)\neq 1$ as the following examples show.

Example 4.8**.**

Let $k=l=6$ and $r=6.$ We have

[TABLE]

Example 4.9**.**

Let $k=20,l=10$ and $r=10.$ We have

[TABLE]

[TABLE]

Remark 4.10**.**

Symmetries in the numbers $T_{k,l}^{j}$ in Examples 4.8 and 4.9 give us some information about orbits of the action of the group $({\mathbb{Z}}/l{\mathbb{Z}})*$ (cf. (3.31) and (3.33)). In Example 4.8 $({\mathbb{Z}}/6{\mathbb{Z}})^{*}\cong{\mathbb{Z}}/2{\mathbb{Z}}$ and we have $80+76$ orbits of length $1$ and $75+78$ orbits of length $2.$ Similarly, in Example 4.9 $({\mathbb{Z}}/10{\mathbb{Z}})^{*}\cong{\mathbb{Z}}/4{\mathbb{Z}}$ and we see that we have $9252+9226$ orbits of length $1$ and $9925+9250$ orbits of length $4.$ The fact that $T_{k,l-1}^{j}=T_{k,l-1}^{l-j}$, which can easily be justified by considering complement Young diagrams or symmetry in the formula (2.18).

Examples 4.8 and 4.9 show that when $\gcd(k,l)\neq 1$ the numbers $T_{k,l}^{j}$ might significantly differ, although one expects the natural densities to be equal once we take their limits as $k,l\rightarrow\infty.$ We can compute only some cases where $\gcd(k,l)\neq 1.$ Let $p$ be an odd prime number. We are able to compute the numbers $T_{Mp,p-1}^{j}$ for $j=0,1,\dots,p-1.$ We start with a key lemma which is a consequence of the action of an additive group ${\mathbb{Z}}/p{\mathbb{Z}}$ on some subsets of restricted partitions.

From now on, to ease the notation, we shall skip the superscript $p$ and write $T_{k,l}^{i}$ instead of ${{}^{p}}T_{k,l}^{i}.$

Define

[TABLE]

Of course, we have the following equalities

[TABLE]

for any $i\in({\mathbb{Z}}/p{\mathbb{Z}})^{*}$ and

[TABLE]

Lemma 4.11**.**

The following equalities hold

[TABLE]

[TABLE]

[TABLE]

Proof..

(4.39) follows from definition. We will use descent with respect to $k.$ We assume inductively that $|S_{m,p-1}^{i}|=|S_{m,p-1}^{j}|$ for any $2\leq m<k$ and $i,j\in{\mathbb{Z}}/p{\mathbb{Z}}.$ To start the induction we see that a map

[TABLE]

where

[TABLE]

and

[TABLE]

is a bijection. But $|V_{2}|=|S_{1,p-1}^{1}|$ which is easily seen by means of the assignment $<p-1,{\pi}_{2}>\,\rightarrow\,<{\pi}_{2}>.$ Similarly, the assignment $<{\pi}_{1},1>\,\rightarrow\,m<{\pi}_{1}>$ yields $|W_{2}|=|S_{1,p-1}^{1}|.$ We can now repeat the above argument for $S_{2,p-1}^{2}$ and $S_{2,p-1}^{4}$ instead of $S^{0}_{2,p-1}$ and $S^{2}_{2,p-1}$ etc. Since $2$ (in fact any $k\neq 0$ ) is a generator of the additive group ${\mathbb{Z}}/p{\mathbb{Z}}$ we see that $|S^{0}_{2,p-1}|=|S^{j}_{2,p-1}|$ for any $j\in({\mathbb{Z}}/p{\mathbb{Z}})^{*}.$

For any $k>2,$ we descent from the statement $|S_{k,p-1}^{0}|=|S_{k,p-1}^{k}|$ to $|S_{k-1,p-1}^{1}|=|S_{k-1,p-1}^{k-1}|.$ This descent is done similarly as for the case $k=2$ by assigning to any partition

[TABLE]

the partition

[TABLE]

But (cf. Fig.2)

[TABLE]

and by inductive hypotheses we get $|S_{k,p-1}^{0}|=|S_{k,p-1}^{k}|.$

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Fig.2

Again, since $k$ is a generator of ${\mathbb{Z}}/p{\mathbb{Z}}$ we see that $|S_{k,p-1}^{0}|=|S_{k,p-1}^{j}|$ for any $j\in({\mathbb{Z}}/p{\mathbb{Z}})^{*}.$ For proving (4.41) we use essentially the same inductive argument. The difference is that this time we have $|S_{1,Mp}^{i}|=|S_{1,Mp}^{j}|$ for any $i,j\in{\mathbb{Z}}/p{\mathbb{Z}}$ and therefore we can start induction from $m=1$. ∎

Remark 4.12**.**

In Fig.2 we depicted the descent process for $k=3$ and $p=5.$ The first diagram describes the assignment (4.46) to any partition of the form (4.45). The second diagram depicts an element from $V_{3}$ and the third corresponding to it an element of $S_{2,4}^{1}$. Similarly, in the fourth diagram we have an element from the set $W_{3}$ and in the fifth corresponding to it an element from $S_{2,4}^{2}.$

Now, we are ready to prove

Theorem 4.13**.**

For any prime number $p,$ any natural $M$ any $1\leq N\leq p-1$ and any $1\leq j\leq p-1$we have the following equalities

[TABLE]

[TABLE]

For any prime $p$ any $1\leq N\leq p-1$ and any $0\leq j\leq p-1$ we have the following equalities

[TABLE]

Proof..

The left hand side of (4.48) is equal to $T_{Mp,N}^{j}$ whereas the left hand side of (4.49) equals $T_{Mp,N}^{0}.$ Notice that for $1\leq j\leq p-1$ we have

[TABLE]

and for $j=0$

[TABLE]

(4.48) and (4.49) follow now from (4.41) of Lemma 4.11 and Corollary 3.25. For $1\leq j\leq p-1$ we have

[TABLE]

and

[TABLE]

By (4.39) and (4.40) we see now that $|T_{p-1,N}^{j}|=|T_{p-1,N}^{0}|$ and (4.50) follows.

∎

Remark 4.14**.**

Notice that formulas (4.48 ) resp. (4.49) are the sums of coefficients ${\sum}_{{n\equiv r}({\text{mod}}p)}a_{n}$ over the nonzero resp. zero moduli classes modulo $p$ of the Gaussian polynomial $G(Mp,N,q)={\sum}_{n=0}^{MpN}a_{n}q^{n}.$ Similarly, (4.50) is the sum over a moduli class mod $p$ of the coefficients of the Gaussian polynomial $G(p-1,N,q)={\sum}_{n=0}^{(p-1)N}a_{n}q^{n}.$

Remark 4.15**.**

Example 4.8 shows that one can not generalise, in a straightforward way,Theorem 4.13 to the case $k=Mp,l=Np-1.$ Take $M=N=2,p=3$ and look at the values appearing in Example 4.8.

We state the following problem

Problem 4.16**.**

Find relations between sums over moduli classes modulo $d$ of coefficients of Gaussian polynomials $G(k,l,q)$ for arbitrary natural numbers $k,l$ and $d.$