A skew Specht perspective of RoCK blocks and cuspidal systems for KLR algebras in affine type A

Cuspidal systems parameterize Khovanov–Lauda–Rouquier (KLR) algebra representations via root partitions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π</mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where simple modules <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis pi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>π</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(\pi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> arise as heads of proper standard modules. Working in affine type A with an arbitrary convex preorder, we construct explicit skew diagrams <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta left-parenthesis pi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>π</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\zeta (\pi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the skew Specht module <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript zeta left-parenthesis pi right-parenthesis"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ζ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>π</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">S^{\zeta (\pi )}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has simple head <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis pi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>π</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(\pi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a filtration by proper standard modules. A key ingredient in this construction is the development of ‘core-truncation’ functors, which take module categories of level one RoCK blocks to the category of imaginary semicuspidal KLR modules. Every simple imaginary semicuspidal module arises in the image of these functors. This result stems from an in-depth study of the combinatorial interplay between cuspidal systems and RoCK cyclotomic KLR algebras, in which we characterize core blocks and RoCK blocks in arbitrary level via cuspidal tiling properties of multipartitions in these blocks.