Another proof of the Nowicki conjecture

TL;DR

This paper provides a new proof of the Nowicki conjecture, which describes the generators of the algebra of constants for a specific derivation on a polynomial algebra, using representation theory of GL_2(K).

Contribution

It offers an alternative proof of the Nowicki conjecture leveraging representation theory, expanding the methods used in previous proofs.

Findings

Confirmed that the algebra of constants is generated by specified elements.

Demonstrated the effectiveness of representation theory in invariant algebra problems.

Provided a new perspective on the structure of derivation invariants.

Abstract

Let $K[X_d,Y_d]=K[x_1,\ldots,x_d,y_1,\ldots,y_d]$ be the polynomial algebra in $2d$ variables over a field $K$ of characteristic 0 and let $\delta$ be the derivation of $K[X_d,Y_d]$ defined by $\delta(y_i)=x_i$, $\delta(x_i)=0$, $i=1,\ldots,d$. In 1994 Nowicki conjectured that the algebra $K[X_d,Y_d]^{\delta}$ of constants of $\delta$ is generated by $X_d$ and $x_iy_j-y_ix_j$ for all $1\leq i<j\leq d$. The affirmative answer was given by several authors using different ideas. In the present paper we give another proof of the conjecture based on representation theory of the general linear group $GL_2(K)$.

Equations

$$\delta (a_1{a}_2)=\delta (a_1)a_2+a_1\delta (a_2),a_1,a_2\in A.$$

$$\delta (x_i)=0,\delta (y_i)=x_i,i=1,\dots ,d,$$

$$u_{ij}=\left|\begin{array}{cc}{x}_i& x_j\\ y_i& y_j\end{array}\right|=x_i{y}_j-x_j{y}_i,1\le i<j\le d.$$

$$g(x)={\gamma }_{11}x+{\gamma }_{21}y,g(y)={\gamma }_{12}x+{\gamma }_{22}y,\text{ where }g=\left(\begin{array}{cc}{\gamma }_{11}& {\gamma }_{12}\\ {\gamma }_{21}& {\gamma }_{22}\end{array}\right)\in G{L}_2(K).$$

$$g(v_1\otimes \cdots \otimes v_n)=g(v_1)\otimes \cdots \otimes g(v_n),v_i\in V,g\in G{L}_2(K).$$

$$\{w_0,w_1,\dots ,w_{{\lambda }_1-{\lambda }_2}\},\mathrm{deg}{w}_i=({\lambda }_1-i,{\lambda }_2+i),i=0,1,\dots ,{\lambda }_1-{\lambda }_2.$$

$$w(\lambda {)}^{(1)}=\underset{{\lambda }_2\text{ times}}{\underbrace{(x\otimes y-y\otimes x)\otimes \cdots \otimes (x\otimes y-y\otimes x)}}\otimes \underset{{\lambda }_1-{\lambda }_2\text{ times}}{\underbrace{x\otimes \cdots \otimes x}}.$$

$$(v_1\otimes \cdots \otimes v_n){\sigma }^{-1}=v_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (n)},v_i\in V,\sigma \in S_n.$$

$$w(\lambda )=w(\lambda {)}^{(1)}\sum _{\sigma \in S_n}{\alpha }_{\sigma }{\sigma }^{-1},{\alpha }_{\sigma }\in K.$$

$$(\sigma (1),\sigma (2)),(\sigma (3),\sigma (4)),\dots ,(\sigma (2{\lambda }_2-1),\sigma (2{\lambda }_2)).$$

$$\sigma (1)<\sigma (2),\sigma (3)<\sigma (4),\dots ,\sigma (2{\lambda }_2-1)<\sigma (2{\lambda }_2),$$

$$\sigma (1)<\sigma (3)<\cdots <\sigma (2{\lambda }_2-1)<\sigma (2{\lambda }_2+1)<\sigma (2{\lambda }_2+2)<\cdots <\sigma (n),$$

$$\sigma (2)<\sigma (4)<\cdots <\sigma (2{\lambda }_2).$$

$$T(V)=\underset{n\ge 0}{⨁}{V}^{{\otimes }_n}=K\oplus V\oplus (V\otimes V)\oplus (V\otimes V\otimes V)\oplus \cdots$$

$$(x_{i_1}\otimes y_{j_1}-y_{i_1}\otimes x_{j_1})\otimes \cdots \otimes (x_{i_{{\lambda }_2}}\otimes y_{j_{{\lambda }_2}}-y_{i_{{\lambda }_2}}\otimes x_{j_{{\lambda }_2}})\otimes x_{k_1}\otimes \cdots \otimes x_{k_{{\lambda }_1-{\lambda }_2}}.$$

$$X_d^p{U}_d^q=x_1^{p_1}\cdots x_d^{p_d}\prod _{1\le i<j\le d}{u}_{ij}^{q_{ij}}$$

$$\{i_a,j_a,k_b\mid a=1,2,\dots ,{\lambda }_2,b=1,2,\dots ,{\lambda }_1-{\lambda }_2\}=\{1,2,\dots ,|n|\}.$$

$$u_{i_1{j}_1}\cdots u_{i_{{\lambda }_2},j_{{\lambda }_2}}{x}_{k_1}\cdots x_{k_{{\lambda }_1-{\lambda }_2}}$$

Full text

1. Introduction

The linear operator $\delta$ of an algebra $A$ over a field $K$ is a derivation if it satisfies the Leibniz rule

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The kernel of $\delta$ is called the algebra of constants of $\delta$ and is denoted by $A^{\delta}$. In the sequel $K$ is a field of characteristic 0. When $A=K[Z_{m}]=K[z_{1},\ldots,z_{m}]$ is the algebra of polynomials in $m$ variables the derivation $\delta$ is called Weitzenböck if it acts as a nilpotent linear operator on the vector space $KZ_{m}$ with basis $Z_{m}=\{z_{1},\ldots,z_{m}\}$. The classical theorem of Weitzenböck [17] in 1932 states that in this case $K[Z_{m}]^{\delta}$ is finitely generated. The algebra $K[Z_{m}]^{\delta}$ coincides with the algebra of invariants $K[Z_{m}]^{(K,+)}$, where the additive group $(K,+)$ of the field $K$ is embedded as a subgroup into the unitriangular group $UT_{m}(K)$ acting as $\{\exp(\alpha\delta)\mid\alpha\in K\}$. Hence the finitely generation of $K[Z_{m}]^{\delta}$ is equivalent to a theorem of classical invariant theory. A modern geometric proof of the Weitzenböck theorem in this spirit is given by Seshadri [15]. A translation in an algebraic language of this proof is given by Tyc [16]. For more information on Weitzenböck derivations one can see the books by Nowicki [14, Section 6.2], Derksen and Kemper [3, Chapter 2], and Dolgachev [4, Section 4.2].

In the special case of the polynomial algebra $K[X_{d},Y_{d}]$ in $2d$ variables $X_{d}=\{x_{1},\ldots,x_{d}\}$ and $Y_{d}=\{y_{1},\ldots,y_{d}\}$ when the Weitzenböck derivation $\delta$ acts by

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Nowicki [14] conjectured in 1994 that $K[X_{d},Y_{d}]^{\delta}$ is generated by $X_{d}$ and the determinants

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There are several proofs based on different methods confirming the Nowicki conjecture: by Khoury [10, 11], Bedratyuk [1], the author and Makar-Limanov [6], Kuroda [12]. There are unpublished proofs by Derksen and Panyushev. As Kuoda mentions in his paper [12] Goto, Hayasaka, Kurano, and Nakamura [8] and Miyazaki [13] determined sets of generators for algebras of invariants with $K[X_{d},Y_{d}]^{\delta}$ included in the list.

In the present paper we give a new proof using easy arguments from representation theory of the general linear group $GL_{2}(K)$. Our proof is inspired by our paper with Gupta [5] devoted to the noncommutative version of Weitzenböck derivations.

2. Preliminaries

Let $V$ be a vector space with basis $\{x,y\}$ with the canonical action of the general linear group $GL_{2}(K)$:

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For a background on representation theory of the general linear group $GL_{m}(K)$ see, e.g., the books by Weyl [18, Chapter 4] or James and Kerber [9, Chapter 8]. We shall summarize the necessary facts for $m=2$ only. The polynomial representations of $GL_{2}(K)$ are completely reducible and their irreducible components $W(\lambda)$ are indexed by partitions $\lambda=(\lambda_{1},\lambda_{2})$. If $\lambda$ is a partition of $n$ (notation $\lambda\vdash n$), then $W(\lambda)$ can be realized as a $GL_{2}(K)$-submodule of the $n$-th tensor degree $V^{\otimes_{n}}$ equipped with the diagonal action of $GL_{2}(K)$

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As a vector space $V^{\otimes_{n}}$ is ${\mathbb{N}}_{0}^{2}$-graded and the homogeneous component of degree $(n_{1},n_{2})$, $n_{1}+n_{2}=n$, is spanned on the tensors $z_{i_{1}}\otimes\cdots\otimes z_{i_{n}}$, $z_{i}=x,y$, of degree $n_{1}$ and $n_{2}$ in $x$ and $y$, respectively. Then $W(\lambda)$ has a basis of homogeneous elements

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The element $w_{0}=w(\lambda)\in W(\lambda)$ which is homogeneous of degree $\lambda$ is called the highest weight vector of $W(\lambda)$. One typical element $w(\lambda)^{(1)}\in W(\lambda)^{(1)}\subset V^{\otimes_{n}}$, $W(\lambda)^{(1)}\cong W(\lambda)$, is

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Here the skew-symmetric sums $(x\otimes y-y\otimes x)$ appear in positions $(1,2),(3,4),\ldots$, $(2\lambda_{2}-1,2\lambda_{2})$. The symmetric group $S_{n}$ acts from the right on $V^{\otimes_{n}}$ by place permutation

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Then every highest weight vector $w(\lambda)\in W(\lambda)\subset V^{\otimes_{n}}$ is of the form

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Clearly, the skew-symmetries in $w(\lambda)^{(1)}\sigma$ are in positions

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Remark 2.1**.**

Since $W(\lambda)$, $\lambda\vdash n$, participates in $V^{\otimes_{n}}$ with multiplicity equal to the number of standard tableaux of shape $[\lambda]$, by [7, Proposition 0.1] we may choose a basis of the vector space of highest weight vectors $w(\lambda)\in V^{\otimes_{n}}$ consisting of all $w(\lambda)^{(1)}\sigma^{-1}$ such that the tableau

\ytableausetup

mathmode, boxsize=5em {ytableau}σ(1)&σ(3)⋯σ(2λ_2-1)σ(2λ_2+1)⋯σ(n)

σ(2)σ(4)⋯σ(2λ_2)

is standard, i.e.,

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The highest weight vector of $W(\lambda)\subset V^{\otimes_{n}}$ can be characterized in the following way, see [2, Lemma 1.1].

Lemma 2.2**.**

Let $\Delta$ be the derivation of the tensor algebra

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defined by $\Delta(x)=0$, $\Delta(y)=x$. Then the homogeneous element $w\not=0$ of degree $\lambda\vdash n$ is a highest weight vector of some $W(\lambda)\subset V^{\otimes_{n}}$ if and only if $\Delta(w)=0$.

Remark 2.3**.**

Up to a nonzero multiplicative constant the derivation $\Delta$ sends the element $w_{i}$ from (4) to $w_{i-1}$, $i=1,\ldots,\lambda_{1}-\lambda_{2}$, and $\Delta(w_{0})=0$.

Let $n=(n_{1},\ldots,n_{d})$ be a $d$-tuple of nonnegative integers and let $|n|=n_{1}+\cdots+n_{d}$. Consider the vector spaces $V_{1},\ldots,V_{d}$ with bases $\{x_{1},y_{1}\},\ldots,\{x_{d},y_{d}\}$, respectively, and the canonical action of $GL_{2}(K)$ as in (3) on them. Clearly, the tensor products $V^{(n)}=V_{1}^{\otimes_{n_{1}}}\otimes\cdots\otimes V_{d}^{\otimes_{n_{d}}}$ and $V^{\otimes_{|n|}}$ are isomorphic as $GL_{2}(K)$-modules. As in the case of $V^{\otimes_{n}}\subset T(V)$ we define an ${\mathbb{N}}_{0}^{2}$-grading on $V^{(n)}$ which counts the number of entries of $X_{d}$ and $Y_{d}$, respectively. Let $\lambda\vdash|n|$. If at the first $\lambda_{2}$ couples of positions $(1,2),(3,4),\ldots,(2\lambda_{2}-1,2\lambda_{2})$ in the tensor product $V^{(n)}$ we have $(V_{i_{1}},V_{j_{1}}),(V_{i_{2}},V_{j_{2}}),\ldots,(V_{i_{\lambda_{2}}},V_{j_{\lambda_{2}}})$, and the positions left are $k_{1},\ldots,k_{\lambda_{1}-\lambda_{2}}$, then the analogue of the equation (5) is

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The equation (6) also can be restated in a similar way. As a consequence of Lemma 2.2 we obtain:

Corollary 2.4**.**

Let $\delta$ be the derivation of the tensor algebra $T(V_{1}\oplus\cdots\oplus V_{d})$ defined by (1). Then a homogeneous element $w\in V^{(n)}=V_{1}^{\otimes_{n_{1}}}\otimes\cdots\otimes V_{d}^{\otimes_{n_{d}}}$ of degree $\lambda\vdash|n|$ is a highest weight vector of a submodule $W(\lambda)$ of $V^{(n)}$ if and only if $\delta(w)=0$.

3. The main result

We are ready to present our proof of the Nowicki conjecture.

Theorem 3.1**.**

Let $K$ be a field of characteristic [math] and let $\delta$ be the derivation of the polynomial algebra $K[X_{d},Y_{d}]$ defined by (1). Then the algebra of constants $K[X_{d},Y_{d}]^{\delta}$ is generated by $X_{d}$ and the determinants (2).

Proof.

The algebra $K[X_{d},Y_{d}]$ has a canonical ${\mathbb{N}}_{0}^{d}$-grading. The homogeneous component $K[X_{d},Y_{d}]^{(n)}$ of degree $n=(n_{1},\ldots,n_{d})$ is spanned by the monomials which are of degree $n_{i}$ in $x_{i}$ and $y_{i}$, $i=1,\ldots,d$. It follows from the definition of $\delta$ that $\delta(K[X_{d},Y_{d}]^{(n)})\subset K[X_{d},Y_{d}]^{(n)}$. Hence we shall prove the theorem if we show that each component $(K[X_{d},Y_{d}]^{(n)})^{\delta}$ is spanned on the products

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of degree $n$. As a $GL_{2}(K)$-module $K[X_{d},Y_{d}]^{(n)}$ is isomorphic to the symmetric tensor power of $n_{1}$ copies of $V_{1}$, $n_{2}$-copies of $V_{2}$, $\ldots$, $n_{d}$ copies of $V_{d}$. Hence it is a homomorphic image of $V^{(n)}=V_{1}^{\otimes_{n_{1}}}\otimes\cdots\otimes V_{d}^{\otimes_{n_{d}}}$. The action of $\delta$ on $V^{(n)}$ induces the canonical action on $K[X_{d},Y_{d}]^{(n)}$. Therefore the vector space of the highest weight vectors of $K[X_{d},Y_{d}]^{(n)}$ is an image of the vector space of the highest weight vectors of $V^{(n)}$ and by Lemma 2.2 and Remark 2.3 coincides with $(K[X_{d},Y_{d}]^{(n)})^{\delta}$. The highest weight vectors of $V^{(n)}$ are linear combinations of the products (7) with the property that

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Obviously the image of the element (7) in $K[X_{d},Y_{d}]^{(n)}$ is

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Replacing $u_{i_{a}j_{a}}$ with $u_{j_{a}i_{a}}$ if $i_{a}>j_{a}$, we obtain that $(K[X_{d},Y_{d}]^{(n)})^{\delta}$ is spanned on the products (8) which completes the proof. ∎