Rank and border rank of Kronecker powers of tensors and Strassen's laser method

<jats:p>We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor <jats:inline-formula><jats:alternatives><jats:tex-math>$$T_{cw,q}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> is the square of its border rank for <jats:inline-formula><jats:alternatives><jats:tex-math>$$q > 2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and that the border rank of its Kronecker cube is the cube of its border rank for <jats:inline-formula><jats:alternatives><jats:tex-math>$$q > 4$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. This answers questions raised implicitly by Coppersmith & Winograd (1990, §11) and explicitly by Bläser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range.</jats:p><jats:p>In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen's laser method, introducing a skew-symmetric version of the Coppersmith–Winograd tensor, <jats:inline-formula><jats:alternatives><jats:tex-math>$$T_{skewcw,q}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mi>k</mml:mi> <mml:mi>e</mml:mi> <mml:mi>w</mml:mi> <mml:mi>c</mml:mi> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>. For <jats:inline-formula><jats:alternatives><jats:tex-math>$$q = 2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, the Kronecker square of this tensor coincides with the <jats:inline-formula><jats:alternatives><jats:tex-math>$$3\times 3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>×</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> determinant polynomial, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\det_{3} \in \mathbb{C}^{9} \otimes \mathbb{C}^{9} \otimes \mathbb{C}^{9}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>det</mml:mo> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>9</mml:mn> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>9</mml:mn> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>9</mml:mn> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two.</jats:p><jats:p>We determine new upper bounds for the (Waring) rank and the (Waring) border rank of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\det_3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>det</mml:mo> <mml:mn>3</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>, exhibiting a strict submultiplicative behaviour for <jats:inline-formula><jats:alternatives><jats:tex-math>$$T_{skewcw,2}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mi>k</mml:mi> <mml:mi>e</mml:mi> <mml:mi>w</mml:mi> <mml:mi>c</mml:mi> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> which is promising for the laser method.</jats:p><jats:p>We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb{C}^{3} \otimes \mathbb{C}^{3} \otimes \mathbb{C}^{3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>

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