Please check these useful links

http://www.sciencedirect.com/science/article/pii/S0944501313001432

http://www.sciencedirect.com/science/article/pii/S2210784316301346

http://link.springer.com/chapter/10.1007/978-81-322-2776-2_21

http://link.springer.com/article/10.1007/s11104-008-9805-z#page-1

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mainly by weathering minerals through acidic exudates.

Potassium solubilisation takes place through  organic acids ( mainly oxalic acid , ferulic acid etc )  released by bacteria as well as plant roots as exudates acted upon insoluble K-bearing minerals.

potassium solubilizing bacteria convert insoluble potassium to soluble form for plants :

first : by acidification trhough the production of organique acide , its the same thing pronunced by Dr.Anoop Kumar Srivastava.

second : by chelation of silicate and aluminium and releasing soluble forme of potassium.